{"id":9745,"date":"2026-07-25T20:25:41","date_gmt":"2026-07-25T11:25:41","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9745"},"modified":"2026-07-27T15:53:35","modified_gmt":"2026-07-27T06:53:35","slug":"math-abstraction-06-vector-spaces","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/mathematical-abstraction\/math-abstraction-06-vector-spaces\/","title":{"rendered":"\ubca1\ud130\uacf5\uac04\uacfc \uc120\ud615 \uad6c\uc870"},"content":{"rendered":"<p><!-- \uc218\ud559\uc801 \ucd94\uc0c1\ud654 6\ud3b8: \ubca1\ud130\uacf5\uac04, \uc0ac\ub840\uc758 \ud1b5\ud569 --><\/p>\n<div class=\"math-abs-series\">\n<p><a href=\"..\/math-abstraction-05-mathematical-structures\/\">5\ubd80<\/a>\uc758 \ub9d0\ubbf8\uc5d0\uc11c \uc608\uace0\ud588\ub4ef, \uc774\uc81c \uc2dc\uc120\uc744 \uad6c\uccb4\uc801\uc778 \uc218\ud559 \uac15\uc758\uc2e4\ub85c \uc62e\uaca8 \ubcf4\uc790. \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd, \uae30\ud558\ud559\uc758 \ubc29\ud5a5\ub7c9, \uc218\uc5f4, \ub2e4\ud56d\uc2dd, \ud568\uc218, \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574\uc5d0\uc11c \uc624\ub298\ub0a0 \uc6b0\ub9ac\uac00 \u2018\uc120\ud615\uc131\u2019\uc774\ub77c\uace0 \ubd80\ub974\ub294 \uacc4\uc0b0\uc740 \uc11c\ub85c \ub2e4\ub978 \ub9e5\ub77d\uc5d0\uc11c \uc624\ub7ab\ub3d9\uc548 \uc0ac\uc6a9\ub418\uc5c8\ub2e4. <!-- \uadf8\ub807\ub2e4\uace0 19\uc138\uae30 \ub9d0\uae4c\uc9c0 \uc774 \ubd84\uc57c\ub4e4\uc774 \uc644\uc804\ud788 \uace0\ub9bd\ub418\uc5b4 \uc788\uc5c8\uac70\ub098 \uc544\ubb34\ub3c4 \uacf5\ud1b5\uc810\uc744 \uc778\uc2dd\ud558\uc9c0 \ubabb\ud588\ub2e4\uace0 \ub9d0\ud560 \uc218\ub294 \uc5c6\ub2e4. --> \ud589\ub82c\uc2dd\uacfc \\(n\\)\ucc28\uc6d0 \uae30\ud558\ud559, \uadf8\ub77c\uc2a4\ub9cc\uc758 \ud655\uc7a5\ub860\ucc98\ub7fc \uc5ec\ub7ec \uc120\ud615 \ubb38\uc81c\ub97c \uc787\ub294 \uc2dc\ub3c4\ub3c4 \uc774\ubbf8 \uc9c4\ud589\ub418\uace0 \uc788\uc5c8\ub2e4. \ub2e4\ub9cc \uc774 \ud750\ub984\ub4e4\uc744 \uc720\ud55c\ucc28\uc6d0\uacfc \ubb34\ud55c\ucc28\uc6d0\uc5d0 \ub450\ub8e8 \uc801\uc6a9\ub418\ub294 \ud558\ub098\uc758 \uacf5\ub9ac\uc801 \uc5b8\uc5b4\ub85c \uc870\uc9c1\ud55c \ubca1\ud130\uacf5\uac04 \uc774\ub860\uc740 20\uc138\uae30 \uc804\ubc18\uc5d0 \uc774\ub974\ub7ec\uc11c\uc57c \ub110\ub9ac \uc790\ub9ac \uc7a1\uc558\ub2e4. \uc774 \uae00\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \uc810\uc9c4\uc801\uc778 \ud1b5\ud569 \uacfc\uc815\uacfc, \uc11c\ub85c \ub2e4\ub978 \uc0ac\ub840\ub97c \ud558\ub098\uc758 \uad6c\uc870\ub85c \ub2e4\ub8f0 \ub54c \uc5bb\ub294 \uc218\ud559\uc801 \uc774\uc810\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<a class=\"math-abs-series-ref\" title=\"Dorier, J.-L. (1995). \u201cA General Outline of the Genesis of Vector Space Theory.\u201d Historia Mathematica, 22(3), 227\u2013261.\" href=\"#ref-Dorier1995\">[Dorier1995]<\/a><a class=\"math-abs-series-ref\" title=\"Moore, G. H. (1995). \u201cThe Axiomatization of Linear Algebra: 1875\u20131940.\u201d Historia Mathematica, 22(3), 262\u2013303.\" href=\"#ref-Moore1995\">[Moore1995]<\/a><\/p>\n<h4>\ud769\uc5b4\uc838 \uc788\ub358 \uc0ac\ub840\ub4e4<\/h4>\n<p>\uc624\ub298\ub0a0 \u2018\uac00\uc6b0\uc2a4 \uc18c\uac70\ubc95\u2019\uc774\ub77c\uace0 \ubd80\ub974\ub294 \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd\uc758 \uc18c\uac70 \uc808\ucc28\ub97c \uac00\uc6b0\uc2a4(Carl Friedrich Gauss)\uac00 \ucc98\uc74c \ubc1c\uba85\ud55c \uac83\uc740 \uc544\ub2c8\ub2e4. \uc18c\uac70\ubc95\uc740 \uac00\uc6b0\uc2a4\ubcf4\ub2e4 \ud6e8\uc52c \uc624\ub798\uc804\ubd80\ud130 \uc0ac\uc6a9\ub418\uc5c8\uace0, \uadfc\uc138 \uc720\ub7fd\uc5d0\uc11c\ub3c4 18\uc138\uae30 \ub9d0\uae4c\uc9c0 \ud1b5\uc0c1\uc801\uc778 \ub300\uc218 \uacc4\uc0b0\ubc95\uc73c\ub85c \uc790\ub9ac \uc7a1\uc558\ub2e4. \uac00\uc6b0\uc2a4\ub294 1809\ub144 \ucc9c\uccb4 \uada4\ub3c4\uc640 \ucd5c\uc18c\uc81c\uacf1 \ubb38\uc81c\ub97c \ub2e4\ub8e8\uba74\uc11c \uc815\uaddc\ubc29\uc815\uc2dd\uc5d0 \uc18c\uac70\ubc95\uc744 \uc801\uc6a9\ud558\uace0, \ub300\uce6d\uc801\uc778 \uacc4\uc0b0\uc5d0 \uc54c\ub9de\uc740 \ud45c\uae30\uc640 \ubc18\ubcf5 \uacc4\uc0b0 \ubc29\uc2dd\uc744 \ubc1c\uc804\uc2dc\ucf30\ub2e4. \uc774\ub7ec\ud55c \uc2e4\ubb34\uac00 \ud6c4\ub300\uc758 \uacc4\uc0b0\ubc95\uc5d0 \ud070 \uc601\ud5a5\uc744 \uc8fc\uba74\uc11c \uadf8\uc758 \uc774\ub984\uc774 \uc18c\uac70\ubc95\uacfc \uacb0\ud569\ud588\uc9c0\ub9cc, \u2018Gaussian elimination\u2019\uc774\ub77c\ub294 \uba85\uce6d \uc790\uccb4\ub294 20\uc138\uae30\uc5d0 \uc815\ucc29\ud588\ub2e4. \ub2f9\uc2dc \uc774 \uacc4\uc0b0\uc740 \uc5f0\ub9bd\ubc29\uc815\uc2dd \ud574\ubc95\uacfc \ucd5c\uc18c\uc81c\uacf1 \uacc4\uc0b0\uc774\ub77c\ub294 \uc2e4\uc6a9\uc801\uc778 \ub9e5\ub77d\uc5d0\uc11c \ubc1c\uc804\ud574 \ub098\uac14\ub2e4.<a class=\"math-abs-series-ref\" title=\"Grcar, J. F. (2011). \u201cHow Ordinary Elimination Became Gaussian Elimination.\u201d Historia Mathematica, 38(2), 163\u2013218.\" href=\"#ref-Grcar2011\">[Grcar2011]<\/a><\/p>\n<p>\uae30\ud558\ud559\uc801 \ubca1\ud130\uc758 \uc5ed\uc0ac\uc5d0\ub294 \uc0ac\uc6d0\uc218(quaternion)\uac00 \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud588\ub2e4. \uc0ac\uc6d0\uc218 \\(q=a+bi+cj+dk\\)\uc5d0\uc11c \uc2e4\uc218\ubd80\uac00 \\(0\\)\uc778 \\(bi+cj+dk\\)\ub294 \uc21c\ud5c8\uc218 \uc0ac\uc6d0\uc218\uc774\uba70, \uc88c\ud45c \\((b,c,d)\\)\ub97c \ud1b5\ud574 3\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \ubca1\ud130\uc640 \ub300\uc751\uc2dc\ud0ac \uc218 \uc788\ub2e4. \ud45c\uc900\uc801\uc778 \ubc29\ud5a5 \uaddc\uc57d\uc744 \ud0dd\ud558\uba74 \ub450 \uc21c\ud5c8\uc218 \uc0ac\uc6d0\uc218 \\(u,v\\)\uc758 \uacf1\uc740<br \/>\n\\[<br \/>\nuv=-u\\mathbin{\\cdot}v+u\\times v<br \/>\n\\]<br \/>\n\ucc98\ub7fc \uc2a4\uce7c\ub77c \ubd80\ubd84\uacfc \ubca1\ud130 \ubd80\ubd84\uc73c\ub85c \ubd84\ud574\ub41c\ub2e4. 1880\ub144\ub300\uc5d0 \uc870\uc0ac\uc774\uc5b4 \uc70c\ub7ec\ub4dc \uae41\uc2a4(Josiah Willard Gibbs)\uc640 \uc62c\ub9ac\ubc84 \ud5e4\ube44\uc0ac\uc774\ub4dc(Oliver Heaviside)\ub294 \uc0ac\uc6d0\uc218\uc640 \uadf8\ub77c\uc2a4\ub9cc\uc758 \uc5f0\uad6c\uc5d0\uc11c \uc601\ud5a5\uc744 \ubc1b\ub418 \uc11c\ub85c \uc0c1\ub2f9 \ubd80\ubd84 \ub3c5\ub9bd\uc801\uc73c\ub85c, \ub0b4\uc801\uacfc \uc678\uc801\uc744 \ubcc4\uac1c\uc758 \uc5f0\uc0b0\uc73c\ub85c \ub2e4\ub8e8\ub294 \ud604\ub300 \ubca1\ud130 \ud574\uc11d\uc758 \ud575\uc2ec \uccb4\uacc4\ub97c \ubc1c\uc804\uc2dc\ucf30\ub2e4. <!-- \uc774\ub97c \ub2e8\uc21c\ud788 \u2018\uc0ac\uc6d0\uc218\uc758 \ud5c8\uc218\ubd80\ub9cc \ub5bc\uc5b4 \ub0b8 \ud45c\uae30\ubc95\u2019\uc774\ub77c\uace0 \ub9d0\ud558\uba74 \ub450 \uc0ac\ub78c\uc774 \uc218\ud589\ud55c \uc7ac\uad6c\uc131\uc758 \ubc94\uc704\ub97c \uc9c0\ub098\uce58\uac8c \ucd95\uc18c\ud558\uac8c \ub41c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Crowe, M. J. (1994 [1967]). A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. Dover Publications. Google Books.\" href=\"#ref-Crowe1994\">[Crowe1994]<\/a> --><\/p>\n<p>[\uae41\uc2a4\uc640 \ud5e4\ube44\uc0ac\uc774\ub4dc\uc758 \ubca1\ud130 \ud574\uc11d\uc740 1890\ub144\ub300\uc5d0 \uc0ac\uc6d0\uc218 \uccb4\uacc4 \uc804\uccb4\ub97c \uc720\uc9c0\ud558\ub824\ub358 \uc9c4\uc601\uacfc \uacf5\uac1c\uc801\uc778 \ub17c\uc7c1\uc744 \ubc8c\uc600\ub2e4. \uc774\ub294 \ub2e8\uc21c\ud55c \uae30\ud638 \uc120\ud0dd\uc744 \ub118\uc5b4, \ubb3c\ub9ac\ud559\uacfc \uae30\ud558\ud559\uc5d0 \uc5b4\ub5a4 \ub300\uc218 \uccb4\uacc4\uac00 \ub354 \uc801\ud569\ud55c\uac00\ub97c \ub458\ub7ec\uc2fc \uacbd\uc7c1\uc774\uc5c8\ub2e4.<a class=\"math-abs-series-ref\" title=\"Crowe, M. J. (1994 [1967]). A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. Dover Publications. Google Books.\" href=\"#ref-Crowe1994\">[Crowe1994]<\/a>]<\/p>\n<p>\uae41\uc2a4\uc640 \ud5e4\ube44\uc0ac\uc774\ub4dc\uc758 \ubca1\ud130 \ud574\uc11d\uc740 \uc8fc\ub85c 3\ucc28\uc6d0 \ubb3c\ub9ac \uacf5\uac04\uc744 \uc704\ud55c \uacc4\uc0b0\ubc95\uc774\uc5c8\ub2e4. \uadf8\ub7ec\ub098 \uac19\uc740 \uc2dc\ub300\uc758 \ubaa8\ub4e0 \uc77c\ubc18\ud654\uac00 3\ucc28\uc6d0 \ud654\uc0b4\ud45c\uc5d0\ub9cc \uba38\ubb3c\ub800\ub358 \uac83\uc740 \uc544\ub2c8\ub2e4. \uadf8\ub77c\uc2a4\ub9cc\uc740 \uc774\ubbf8 \uc784\uc758 \ucc28\uc6d0\uc758 \ud655\uc7a5\ub7c9\uc744 \uc5f0\uad6c\ud588\uace0, \uc218\uc5f4, \ub2e4\ud56d\uc2dd, \ud568\uc218\uc5d0\uc11c\ub3c4 \ub367\uc148\uacfc \uc0c1\uc218\ubc30\ub97c \uc774\uc6a9\ud55c \uc120\ud615\uc801 \ubc29\ubc95\uc774 \uac01 \ubd84\uc57c \uc548\uc5d0\uc11c \uc4f0\uc774\uace0 \uc788\uc5c8\ub2e4. \ubbf8\ubd84\ubc29\uc815\uc2dd\uc5d0\uc11c\ub294 \uc870\uac74\uc744 \ub354 \ubd84\uba85\ud788 \ud574\uc57c \ud55c\ub2e4. \uc120\ud615 \ubbf8\ubd84\uc5f0\uc0b0\uc790 \\(L\\)\uc5d0 \ub300\ud55c <em>\ub3d9\ucc28<\/em> \uc120\ud615\ubc29\uc815\uc2dd \\(L[y]=0\\)\uc5d0\uc11c\ub294 \\(L[y_1]=L[y_2]=0\\)\uc77c \ub54c \ubaa8\ub4e0 \uc2a4\uce7c\ub77c \\(\\alpha,\\beta\\)\uc5d0 \ub300\ud558\uc5ec \\(L[\\alpha y_1+\\beta y_2]=0\\)\uc774\ubbc0\ub85c \ud574\uc9d1\ud569\uc774 \ubca1\ud130\uacf5\uac04\uc744 \uc774\ub8ec\ub2e4. <!-- \ubc18\uba74 \\(f\\ne0\\)\uc778 \ube44\uc81c\ucc28 \uc120\ud615\ubc29\uc815\uc2dd \\(L[y]=f\\)\uc758 \ud574\uc9d1\ud569\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \ubca1\ud130\uacf5\uac04\uc774 \uc544\ub2c8\ub77c \ud558\ub098\uc758 \ud2b9\uc218\ud574\uc5d0 \ub3d9\ucc28 \ud574\uacf5\uac04\uc744 \ub354\ud55c \uc544\ud540\uacf5\uac04\uc774\uba70, \ube44\uc120\ud615 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc5d0\ub294 \uc911\ucca9 \uc6d0\ub9ac\uac00 \uc77c\ubc18\uc801\uc73c\ub85c \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4.<a class=\"math-abs-series-ref\" title=\"Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics 140, American Mathematical Society.\" href=\"#ref-Teschl2012\">[Teschl2012]<\/a> --><\/p>\n<h4>\ud398\uc544\ub178\uc758 \uc774\ub978 \uacf5\ub9ac\ud654<\/h4>\n<p>\uc774\ub7ec\ud55c \uc77c\ubc18\ud654\uc758 \uc911\uc694\ud55c \uc774\uc815\ud45c\ub97c \uc138\uc6b4 \uc778\ubb3c\uc740 \uc774\ud0c8\ub9ac\uc544 \uc218\ud559\uc790 \uc8fc\uc138\ud398 \ud398\uc544\ub178(Giuseppe Peano)\uc774\ub2e4. \ud398\uc544\ub178\ub294 1888\ub144 \u201cH. \uadf8\ub77c\uc2a4\ub9cc\uc758 \ud655\uc7a5\ub860\uc5d0 \ub530\ub978 \uae30\ud558 \uacc4\uc0b0, \uc5f0\uc5ed\ub17c\ub9ac\uc758 \uc5f0\uc0b0\uc744 \uc55e\uc138\uc6cc\u201d(<i>Calcolo geometrico secondo l&#8217;Ausdehnungslehre di H. Grassmann, preceduto dalle operazioni della logica deduttiva<\/i>)\ub97c \ucd9c\ud310\ud588\ub2e4. \ucc45\uc758 \ub9c8\uc9c0\ub9c9 \uc7a5\uc5d0\uc11c \uadf8\ub294 \ub300\uc0c1\uc758 \uad6c\uccb4\uc801\uc778 \ubcf8\uc131\uc744 \ubbf8\ub9ac \uc815\ud558\uc9c0 \uc54a\uc740 \u2018\uc120\ud615\uacc4\u2019(<i>sistema lineare<\/i>)\ub97c \uc815\uc758\ud588\ub2e4. \uadf8 \uc6d0\uc18c\ub4e4\uc5d0\ub294 \ub367\uc148\uacfc \uc2e4\uc218\uc5d0 \uc758\ud55c \uc2a4\uce7c\ub77c\ubc30\uac00 \uc8fc\uc5b4\uc9c0\uace0, \uc774 \uc5f0\uc0b0\ub4e4\uc740 \uc624\ub298\ub0a0\uc758 \uc2e4\ubca1\ud130\uacf5\uac04 \uacf5\ub9ac\uc640 \ubcf8\uc9c8\uc801\uc73c\ub85c \uac19\uc740 \ubc95\uce59\uc744 \ub9cc\uc871\ud55c\ub2e4. <!-- \ub2e4\ub9cc \ud398\uc544\ub178\uac00 \uc784\uc758\uc758 \uccb4 \uc704\uc758 \ubca1\ud130\uacf5\uac04\uc744 \uc815\uc758\ud588\ub2e4\uac70\ub098, \uc774 \ucc45\uc5d0\uc11c \uc5f0\ub9bd\ubc29\uc815\uc2dd\u00b7\uc218\uc5f4\u00b7\ub2e4\ud56d\uc2dd\u00b7\ud568\uc218\u00b7\ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574\ub97c \ubaa8\ub450 \ud558\ub098\uc758 \uc5f0\uad6c \ud504\ub85c\uadf8\ub7a8\uc73c\ub85c \uba85\uc2dc\uc801\uc73c\ub85c \ud1b5\ud569\ud588\ub2e4\uace0 \ub9d0\ud574\uc11c\ub294 \uc548 \ub41c\ub2e4. \ucc45\uc758 \uc8fc\ub41c \ub0b4\uc6a9\uacfc \uc801\uc6a9\uc740 \uae30\ud558\ud559\uc774\uc5c8\ub2e4.<a class=\"math-abs-series-ref\" title=\"Peano, G. (1888). Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann, preceduto dalle operazioni della logica deduttiva. Torino: Fratelli Bocca. \uc6d0\ubb38 \uc2a4\uce94.\" href=\"#ref-Peano1888\">[Peano1888]<\/a><a class=\"math-abs-series-ref\" title=\"Dorier, J.-L. (1995). \u201cA General Outline of the Genesis of Vector Space Theory.\u201d Historia Mathematica, 22(3), 227\u2013261.\" href=\"#ref-Dorier1995\">[Dorier1995]<\/a><a class=\"math-abs-series-ref\" title=\"Moore, G. H. (1995). \u201cThe Axiomatization of Linear Algebra: 1875\u20131940.\u201d Historia Mathematica, 22(3), 262\u2013303.\" href=\"#ref-Moore1995\">[Moore1995]<\/a> --><\/p>\n<p>[\uadf8\ub808\uace0\ub9ac H. \ubb34\uc5b4(Gregory H. Moore)\uc758 1995\ub144 \ub17c\ubb38 \u201c\uc120\ud615\ub300\uc218\ud559\uc758 \uacf5\ub9ac\ud654, 1875\u20131940\u201d(<i>The Axiomatization of Linear Algebra: 1875\u20131940<\/i>)\uc740 \uc2e4\uc81c\ub85c \uc874\uc7ac\ud558\uba70, \ud398\uc544\ub178\uc758 1888\ub144 \uc815\uc758\uc5d0\uc11c 20\uc138\uae30 \uc804\ubc18\uc758 \ubca1\ud130\uacf5\uac04 \uc774\ub860\uc5d0 \uc774\ub974\ub294 \uacfc\uc815\uc744 \ucd94\uc801\ud55c \ud45c\uc900\uc801\uc778 \uc218\ud559\uc0ac \uc5f0\uad6c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Moore, G. H. (1995). \u201cThe Axiomatization of Linear Algebra: 1875\u20131940.\u201d Historia Mathematica, 22(3), 262\u2013303.\" href=\"#ref-Moore1995\">[Moore1995]<\/a>]<\/p>\n<p><!-- \ud398\uc544\ub178\uc758 \ucc45\uc740 \uc81c\ubaa9\ub300\ub85c \ud5e4\ub974\ub9cc \uadf8\ub77c\uc2a4\ub9cc(Hermann Grassmann)\uc758 \ud655\uc7a5\ub860\uc744 \ucd9c\ubc1c\uc810\uc73c\ub85c \uc0bc\uc558\uc9c0\ub9cc \ub2e8\uc21c\ud55c \ud574\uc124\uc11c\uc5d0 \uba38\ubb3c\uc9c0\ub294 \uc54a\uc558\ub2e4. -->\uadf8\ub77c\uc2a4\ub9cc\uc740 1844\ub144 \u201c\uc120\ud615 \ud655\uc7a5\ub860\u201d(<i>Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik<\/i>)\uc5d0\uc11c \uc784\uc758 \ucc28\uc6d0\uc744 \uac16\ub294 \uccb4\uacc4, \uc77c\ucc28\ub3c5\ub9bd, \uc0dd\uc131\uacfc \ucc28\uc6d0\uc5d0 \uac00\uae4c\uc6b4 \uac1c\ub150, \uc624\ub298\ub0a0\uc758 \uad50\ud658\uc815\ub9ac\uc5d0 \ud574\ub2f9\ud558\ub294 \uacb0\uacfc\uc640 \uc678\uc801\uc758 \uc804\uc2e0\uc744 \ubc1c\uc804\uc2dc\ucf30\ub2e4. \ud398\uc544\ub178\ub294 \uc774 \uc804\ud1b5\uc744 \ub354 \uba85\uc2dc\uc801\uc778 \uacf5\ub9ac\uc801 \ud615\uc2dd\uc73c\ub85c \uc7ac\uad6c\uc131\ud588\ub2e4.<a class=\"math-abs-series-ref\" title=\"Dorier, J.-L. (1995). \u201cA General Outline of the Genesis of Vector Space Theory.\u201d Historia Mathematica, 22(3), 227\u2013261.\" href=\"#ref-Dorier1995\">[Dorier1995]<\/a><a class=\"math-abs-series-ref\" title=\"Fearnley-Sander, D. (1979). \u201cHermann Grassmann and the Creation of Linear Algebra.\u201d The American Mathematical Monthly, 86(10), 809\u2013817.\" href=\"#ref-FearnleySander1979\">[FearnleySander1979]<\/a><\/p>\n<p>\uadf8\ub77c\uc2a4\ub9cc\uc758 1844\ub144 \uc800\uc220\uc740 \ucca0\ud559\uc801 \uc5b8\uc5b4\uc640 \ub0af\uc120 \ud615\uc2dd, \ub192\uc740 \uc77c\ubc18\uc131 \ub54c\ubb38\uc5d0 \uc77d\uae30 \uc5b4\ub824\uc6e0\uace0 \uc989\uac01\uc801\uc778 \ubc18\uc751\ub3c4 \uc81c\ud55c\uc801\uc774\uc5c8\ub2e4. <!-- \uadf8\ub294 \ub300\ud559 \uad50\uc218\uac00 \uc544\ub2c8\ub77c \uc8fc\ub85c \uae40\ub098\uc9c0\uc6c0\uc5d0\uc11c \uac00\ub974\ucce4\uc9c0\ub9cc, \uadf8\uc758 \uc800\uc220\uc774 \ubc1b\uc544\ub4e4\uc5ec\uc9c0\uc9c0 \uc54a\uc740 \uc6d0\uc778\uc744 \uc2e0\ubd84 \ud558\ub098\ub85c \ud658\uc6d0\ud558\uac70\ub098 \uadf8\ub97c \u2018\uace0\ub4f1\ud559\uad50 \uad50\uc0ac\uc5d0 \ubd88\uacfc\ud588\ub2e4\u2019\uace0 \ud45c\ud604\ud558\ub294 \uac83\uc740 \ubd80\uc815\ud655\ud558\ub2e4. --><!-- \ub9c8\ucc2c\uac00\uc9c0\ub85c \ud398\uc544\ub178\uc758 \uacf5\ub9ac\ud654\uac00 \uadf8\ub77c\uc2a4\ub9cc\uc758 \ucc45\uc5d0 \uc885\uc18d\ub418\uc5b4 \uc788\uc5c8\uae30 \ub54c\ubb38\uc5d0 \ubc18\uc138\uae30 \ub3d9\uc548 \uc644\uc804\ud788 \uc78a\ud614\ub2e4\uace0 \ub2e8\uc815\ud558\uae30\ub3c4 \uc5b4\ub835\ub2e4. \ub354 \uc815\ud655\ud55c \uc124\uba85\uc740 \ud398\uc544\ub178\uc758 \uc815\uc758\uac00 \uc989\uc2dc \ud3ed\ub113\uc740 \uc5f0\uad6c \ud504\ub85c\uadf8\ub7a8\uc73c\ub85c \uacc4\uc2b9\ub418\uc9c0 \uc54a\uc558\uc73c\uba70, 1918\ub144\uc758 \ud5e4\ub974\ub9cc \ubc14\uc77c\uacfc 1920\ub144 \uc804\ud6c4\uc758 \ud574\uc11d\ud559\uc790\u00b7\ub300\uc218\ud559\uc790\ub4e4\uc774 \ube44\uc2b7\ud55c \uac1c\ub150\uc744 \uc0c1\ub2f9 \ubd80\ubd84 \ub3c5\ub9bd\uc801\uc73c\ub85c \ub2e4\uc2dc \uc815\uc2dd\ud654\ud588\ub2e4\ub294 \uac83\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Dorier, J.-L. (1995). \u201cA General Outline of the Genesis of Vector Space Theory.\u201d Historia Mathematica, 22(3), 227\u2013261.\" href=\"#ref-Dorier1995\">[Dorier1995]<\/a><a class=\"math-abs-series-ref\" title=\"Moore, G. H. (1995). \u201cThe Axiomatization of Linear Algebra: 1875\u20131940.\u201d Historia Mathematica, 22(3), 262\u2013303.\" href=\"#ref-Moore1995\">[Moore1995]<\/a><a class=\"math-abs-series-ref\" title=\"Fearnley-Sander, D. (1979). \u201cHermann Grassmann and the Creation of Linear Algebra.\u201d The American Mathematical Monthly, 86(10), 809\u2013817.\" href=\"#ref-FearnleySander1979\">[FearnleySander1979]<\/a> --><\/p>\n<h4>\ubc14\ub098\ud750\uc640 \ud568\uc218\ud574\uc11d<\/h4>\n<p>20\uc138\uae30 \ucd08 \uc801\ubd84\ubc29\uc815\uc2dd\uacfc \ubb34\ud55c\uae09\uc218, \ud568\uc218\uc5f4\uc744 \uc5f0\uad6c\ud558\ub358 \ud574\uc11d\ud559\uc790\ub4e4\uc740 \ud568\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc744 \ud558\ub098\uc758 \uacf5\uac04\uc73c\ub85c \ub2e4\ub8e8\uc5b4\uc57c \ud560 \ud544\uc694\ub97c \uac15\ud558\uac8c \ub290\uaf08\ub2e4. \uc774 \ud750\ub984\uc5d0\ub294 \ud790\ubca0\ub974\ud2b8, \ud504\ub808\uc170, \ub9ac\uc2a4\uc758 \uc120\ud589 \uc5f0\uad6c\uac00 \uc788\uc5c8\uace0, 1920\ub144 \uc804\ud6c4\uc5d0\ub294 \ubc14\ub098\ud750, \ud55c, \uc704\ub108\uc640 \ub1cc\ud130 \ub4f1\uc774 \uc11c\ub85c \ub2e4\ub978 \ubb38\uc81c\uc5d0\uc11c \ubca1\ud130\uacf5\uac04\uc5d0 \uac00\uae4c\uc6b4 \uacf5\ub9ac\uc801 \uac1c\ub150\uc744 \ubc1c\uc804\uc2dc\ucf30\ub2e4. <!-- \ub530\ub77c\uc11c \ud604\ub300\uc758 \ubca1\ud130\uacf5\uac04 \uc5b8\uc5b4\uac00 \ud398\uc544\ub178\uc758 \uc800\uc791\uc744 \ub2e8\uc21c\ud788 \uc7ac\ubc1c\uacac\ud55c \uacb0\uacfc\uc774\uac70\ub098, \ud55c \uc0ac\ub78c\uc758 \ub2e8\uc77c\ud55c \ubc1c\uba85\uc73c\ub85c \uc644\uc131\ub418\uc5c8\ub2e4\uace0 \ubcf4\ub294 \uac83\uc740 \uc9c0\ub098\uce58\uac8c \ub2e8\uc21c\ud558\ub2e4.<a class=\"math-abs-series-ref\" title=\"Dorier, J.-L. (1995). \u201cA General Outline of the Genesis of Vector Space Theory.\u201d Historia Mathematica, 22(3), 227\u2013261.\" href=\"#ref-Dorier1995\">[Dorier1995]<\/a><a class=\"math-abs-series-ref\" title=\"Moore, G. H. (1995). \u201cThe Axiomatization of Linear Algebra: 1875\u20131940.\u201d Historia Mathematica, 22(3), 262\u2013303.\" href=\"#ref-Moore1995\">[Moore1995]<\/a> --><\/p>\n<p>\ud3f4\ub780\ub4dc\uc758 \uc218\ud559\uc790 \uc2a4\ud14c\ud310 \ubc14\ub098\ud750(Stefan Banach)\ub294 1920\ub144\uc5d0 \uc81c\ucd9c\ud55c \ubc15\uc0ac\ub17c\ubb38\uc758 \uc131\uacfc\ub97c 1922\ub144 \ub17c\ubb38 \u201c\ucd94\uc0c1 \uc9d1\ud569\uc5d0\uc11c\uc758 \uc5f0\uc0b0\uacfc \uc801\ubd84\ubc29\uc815\uc2dd\uc73c\ub85c\uc758 \uc751\uc6a9\u201d(<i>Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales<\/i>)\uc73c\ub85c \ucd9c\ud310\ud588\ub2e4. \uc774 \ub17c\ubb38\uc5d0\uc11c \uadf8\ub294 \uc624\ub298\ub0a0\uc758 \uc644\ube44 \ub178\ub984 \uc2e4\ubca1\ud130\uacf5\uac04\uc5d0 \ud574\ub2f9\ud558\ub294 \ucd94\uc0c1 \uacf5\uac04\uc744 \uacf5\ub9ac\uc801\uc73c\ub85c \ub2e4\ub8e8\uace0, \uadf8 \uc704\uc758 \uc5f0\uc0b0\uacfc \uc801\ubd84\ubc29\uc815\uc2dd\uc5d0 \uc801\uc6a9\ub418\ub294 \uc77c\ubc18 \uc815\ub9ac\ub4e4\uc744 \uc5f0\uad6c\ud588\ub2e4. \ub300\uc218\uc801\uc778 \ubca1\ud130\uacf5\uac04 \uad6c\uc870\uc5d0 \ub178\ub984\uacfc \uc644\ube44\uc131\uc744 \uacb0\ud569\ud568\uc73c\ub85c\uc368 \ud568\uc218\uc640 \uc218\uc5f4\uc758 \uacf5\uac04\uc744 \uacf5\ud1b5\ub41c \ud574\uc11d\ud559\uc801 \ub300\uc0c1\uc73c\ub85c \ub2e4\ub8f0 \uc218 \uc788\ub294 \ud2c0\uc774 \uc120\uba85\ud574\uc84c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Banach, S. (1922). \u201cSur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales.\u201d Fundamenta Mathematicae, 3, 133\u2013181. \uc6d0\ubb38: EuDML.\" href=\"#ref-Banach1922\">[Banach1922]<\/a><a class=\"math-abs-series-ref\" title=\"Pietsch, A. (2007). History of Banach Spaces and Linear Operators. Birkh\u00e4user.\" href=\"#ref-Pietsch2007\">[Pietsch2007]<\/a><\/p>\n<p>[\u2018\ubc14\ub098\ud750\uacf5\uac04\u2019\uc774\ub77c\ub294 \uba85\uce6d\uc740 \ubc14\ub098\ud750 \uc790\uc2e0\uc758 1922\ub144 \ub17c\ubb38 \uc81c\ubaa9\uc5d0 \ub4f1\uc7a5\ud558\uc9c0 \uc54a\ub294\ub2e4. \ubb38\ud5cc\uc0ac\ub97c \uc815\ub9ac\ud55c \ud53c\ucde8\uc5d0 \ub530\ub974\uba74 \ud504\ub808\uc170\uac00 1928\ub144\uc5d0 \u2018\ubc14\ub098\ud750 \uc528\uc758 \uacf5\uac04\ub4e4\u2019\uc774\ub77c\ub294 \ud45c\ud604\uc744 \uc0ac\uc6a9\ud588\uace0, \uadf8 \ub4a4 \uc774 \uba85\uce6d\uc774 \uc815\ucc29\ud588\ub2e4.<a class=\"math-abs-series-ref\" title=\"Pietsch, A. (2007). History of Banach Spaces and Linear Operators. Birkh\u00e4user.\" href=\"#ref-Pietsch2007\">[Pietsch2007]<\/a>]<\/p>\n<h4>\ubca1\ud130\uacf5\uac04\uc758 \uc815\uc758\uc640 \uadf8 \uc0ac\ub840<\/h4>\n<div class=\"box theorem\">\n<p>\n<span class=\"theorem\">\uc815\uc758: \ubca1\ud130\uacf5\uac04<\/span><br \/>\n\uccb4(field) \\(F\\) \uc704\uc758 \uc9d1\ud569 \\(V\\)\uc5d0 \ub450 \uc5f0\uc0b0, \uc989 \ubca1\ud130 \ub367\uc148 \\(+:V\\times V\\to V\\)\uc640 \uc2a4\uce7c\ub77c\ubc30 \\(\\cdot:F\\times V\\to V\\)\uac00 \uc815\uc758\ub418\uc5b4 \uc788\uace0 \ub2e4\uc74c \uc870\uac74\ub4e4\uc744 \ub9cc\uc871\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n<ul>\n<li>\\(V\\)\ub294 \ub367\uc148\uc5d0 \uad00\ud558\uc5ec \uc544\ubca8\uad70(\uac00\ud658\uad70)\uc774\ub2e4. \uc989 \ub367\uc148\uc740 \uacb0\ud569\ubc95\uce59\uacfc \uad50\ud658\ubc95\uce59\uc744 \ub9cc\uc871\ud558\uace0, \ub367\uc148 \ud56d\ub4f1\uc6d0 \\(0\\in V\\)\uc640 \uac01 \\(v\\in V\\)\uc758 \ub367\uc148 \uc5ed\uc6d0 \\(-v\\in V\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(\\alpha,\\beta\\in F\\)\uc640 \\(v\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\alpha(\\beta v)=(\\alpha\\beta)v\\)\uc774\uace0 \\(1v=v\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(\\alpha,\\beta\\in F\\)\uc640 \\(v,w\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\alpha(v+w)=\\alpha v+\\alpha w\\) \ubc0f \\((\\alpha+\\beta)v=\\alpha v+\\beta v\\)\uc774\ub2e4.<\/li>\n<\/ul>\n<p>\uc774 \uc870\uac74\ub4e4\uc744 \ubaa8\ub450 \ub9cc\uc871\ud558\uba74 \\(V\\)\ub97c \uccb4 \\(F\\) \uc704\uc758 <span class=\"defined\">\ubca1\ud130\uacf5\uac04<\/span>(vector space)\uc774\ub77c \ud558\uace0, \\(V\\)\uc758 \uc6d0\uc18c\ub97c <span class=\"defined\">\ubca1\ud130<\/span>(vector)\ub77c\uace0 \ud55c\ub2e4.<\/p>\n<\/div>\n<p>\uc774 \uc815\uc758\ub97c \ub9cc\uc871\ud558\ub294 \ub300\ud45c\uc801\uc778 \uc0ac\ub840\ub4e4\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<ul>\n<li>\uccb4 \\(F\\)\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \\(n\\)-\ud29c\ud50c \uacf5\uac04 \\(F^n\\) (\uc131\ubd84\ubcc4 \ub367\uc148\uacfc \uc2a4\uce7c\ub77c\ubc30)<\/li>\n<li>\ucc28\uc218\uac00 \\(n\\) \uc774\ud558\uc778 \ub2e4\ud56d\uc2dd \uacf5\uac04 \\(P_n(F)=\\{a_0+a_1x+\\cdots+a_nx^n:a_i\\in F\\}\\)<\/li>\n<li>\uc218\uc5f4\uacf5\uac04 \\(F^{\\mathbb N}\\) (\uc810\ubcc4 \ub367\uc148\uacfc \uc2a4\uce7c\ub77c\ubc30)<\/li>\n<li>\ub2eb\ud78c\uad6c\uac04 \\([a,b]\\) \uc704\uc758 \uc5f0\uc18d\ud568\uc218 \uacf5\uac04 \\(C([a,b],F)\\) (\uc810\ubcc4 \ub367\uc148\uacfc \uc2a4\uce7c\ub77c\ubc30)<\/li>\n<li>\ub3d9\ucc28 \uc120\ud615 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \uc2e4\uc218\uac12 \ud574\uacf5\uac04 \\(\\{y\\in C^2(\\mathbb R,\\mathbb R):y&#8221;+y=0\\}\\)<\/li>\n<\/ul>\n<p>\uc774 \uc0ac\ub840\ub4e4\uc740 \uc6d0\uc18c\uc758 \uad6c\uccb4\uc801\uc778 \ubaa8\uc2b5\uc774 \uc11c\ub85c \ub2e4\ub974\uc9c0\ub9cc \ubaa8\ub450 \ubca1\ud130\uacf5\uac04\uc774\ub2e4. <!-- \uadf8\ub7ec\ub098 \uadf8\uac83\ub9cc\uc73c\ub85c \uc774 \uacf5\uac04\ub4e4\uc774 \uc11c\ub85c \ub3d9\ud615\uc774 \ub418\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. --> <!-- \ub2e8, \ubca1\ud130\uacf5\uac04\ub4e4\uc740 \uccb4\ub098 \ucc28\uc6d0\uc774 \ub2e4\ub97c \uc218 \uc788\uace0, \ub178\ub984, \ub0b4\uc801, \uacf1\uc148\ucc98\ub7fc \ubcc4\ub3c4\uc758 \uad6c\uc870\ub97c \uc9c0\ub2d0 \uc218\ub3c4 \uc788\ub2e4. --> <!-- \uc815\ud655\ud55c \uacb0\ub860\uc740 \uc624\uc9c1 \ubca1\ud130\uacf5\uac04 \uacf5\ub9ac\uc5d0\uc11c \uc99d\uba85\ub41c \uc815\ub9ac\uac00 \uadf8 \uc815\ub9ac\uc5d0 \uba85\uc2dc\ub41c \ucd94\uac00 \uac00\uc815\uae4c\uc9c0 \ub9cc\uc871\ud558\ub294 \ubaa8\ub4e0 \uc0ac\ub840\uc5d0 \uc801\uc6a9\ub41c\ub2e4\ub294 \uac83\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Axler, S. (2024). Linear Algebra Done Right, 4th ed. Springer.\" href=\"#ref-Axler2024\">[Axler2024]<\/a> --><\/p>\n<h4>\uc65c \ud1b5\ud569\uc774 \uc774\ub4dd\uc778\uac00<\/h4>\n<p>\uc2e4\uc218\uac12 \ud574\ub97c \uc0dd\uac01\ud558\uc5ec<br \/>\n\\[<br \/>\nV=\\{y\\in C^2(\\mathbb R,\\mathbb R):y&#8221;+y=0\\}<br \/>\n\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \\(L:C^2(\\mathbb R,\\mathbb R)\\to C(\\mathbb R,\\mathbb R)\\), \\(L(y)=y&#8221;+y\\)\ub294 \uc120\ud615\uc0ac\uc0c1\uc774\uace0 \\(V=\\ker L\\)\uc774\ubbc0\ub85c \\(V\\)\ub294 \ubca1\ud130\uacf5\uac04\uc774\ub2e4.<\/p>\n<p>[\ubaa8\ub4e0 \\(y\\in V\\)\uac00 \uc720\uc77c\ud558\uac8c \\(y=c_1\\cos x+c_2\\sin x\\)\ub85c \ud45c\ud604\ub41c\ub2e4\ub294 \uc0ac\uc2e4\uc740 \uc0c1\ubbf8\ubd84\ubc29\uc815\uc2dd\ub860\uc758 \ud45c\uc900 \uacb0\uacfc\ub2e4. \ucd08\uae30\uac12 \uc874\uc7ac\u00b7\uc720\uc77c\uc131 \uc815\ub9ac\ub85c\ub3c4 \uc774\ub97c \ubcfc \uc218 \uc788\ub2e4. \ucd08\uae30\uac12 \uc0ac\uc0c1 \\(E:V\\to\\mathbb R^2\\), \\(E(y)=(y(0),y'(0))\\)\ub294 \uc120\ud615\uc774\uace0, \uac01 \\((a,b)\\in\\mathbb R^2\\)\uc5d0 \ub300\ud558\uc5ec \ud574\ub2f9 \ucd08\uae30\uac12\uc744 \uac16\ub294 \ud574\uac00 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4. \uc2e4\uc81c\ub85c \uadf8 \ud574\ub294 \\(a\\cos x+b\\sin x\\)\uc774\ubbc0\ub85c \\(E\\)\ub294 \uc120\ud615\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Graduate Studies in Mathematics 140, American Mathematical Society.\" href=\"#ref-Teschl2012\">[Teschl2012]<\/a>]<\/p>\n<p>\ub450 \ud568\uc218 \\(\\cos x\\)\uc640 \\(\\sin x\\)\ub294 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4. \uc2e4\uc81c\ub85c \\(c_1\\cos x+c_2\\sin x=0\\)\uc774 \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0\uc11c \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uba74, \\(x=0\\)\uc5d0\uc11c \\(c_1=0\\)\uc744 \uc5bb\uace0 \\(x=\\pi\/2\\)\uc5d0\uc11c \\(c_2=0\\)\uc744 \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \\(\\{\\cos x,\\sin x\\}\\)\ub294 \\(V\\)\uc758 \uae30\uc800\uc774\uace0 \\(\\dim V=2\\)\uc774\ub2e4.<\/p>\n<p><!-- \uc774 \uc0ac\uc2e4\uc744 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \uc5b8\uc5b4\ub9cc\uc73c\ub85c\ub3c4 \uac1c\ubcc4\uc801\uc73c\ub85c \uc99d\uba85\ud560 \uc218\ub294 \uc788\ub2e4. -->\ubca1\ud130\uacf5\uac04\uc774\ub77c\ub294 \ucd94\uc0c1 \uc5b8\uc5b4\uc758 \uc774\uc810\uc740 \ud568\uc218\uc758 \ud574\uc9d1\ud569\uc5d0 \ub300\ud55c \u2018\uae30\uc800\u2019\uc640 \u2018\ucc28\uc6d0\u2019\uc744 \uae30\ud558 \ubca1\ud130\uc5d0 \uc0ac\uc6a9\ud558\ub358 \uac83\uacfc \ub3d9\uc77c\ud55c \uc815\uc758\uc640 \uc815\ub9ac\ub85c \ub2e4\ub8f0 \uc218 \uc788\uac8c \ud55c\ub2e4\ub294 \ub370 \uc788\ub2e4. \ub2e4\uc74c \uc808\uc5d0\uc11c\ub294 \uc774 \ucc28\uc6d0\uc744 \uc815\uc758\ud558\uace0, \ucc28\uc6d0\uc774 \uac19\uc740 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\ub4e4\uc774 \uc815\ud655\ud788 \uac19\uc740 \ub300\uc218 \uad6c\uc870\ub97c \uac16\ub294\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc99d\uba85\ud55c\ub2e4.<\/p>\n<h4>\uc720\ud55c\ucc28\uc6d0\uc758 \ubd84\ub958<\/h4>\n<p>\ubca1\ud130\uacf5\uac04 \\(V\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(S\\)\uc5d0 \ub300\ud558\uc5ec, \\(S\\)\uc758 \uc6d0\uc18c\ub4e4\uc744 \uc0ac\uc6a9\ud55c \ubaa8\ub4e0 <em>\uc720\ud55c<\/em> \uc77c\ucc28\uacb0\ud569\uc758 \uc9d1\ud569\uc744 \\(S\\)\uc758 <span class=\"defined\">\uc0dd\uc131\uacf5\uac04<\/span>(span)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\operatorname{span}(S)=V\\)\uc774\uba74 \\(S\\)\uac00 \\(V\\)\ub97c \uc0dd\uc131\ud55c\ub2e4\uace0 \ub9d0\ud55c\ub2e4. \ud55c\ud3b8 \\(S\\)\uc758 \uc11c\ub85c \ub2e4\ub978 \uc720\ud55c \uac1c \uc6d0\uc18c \\(v_1,\\ldots,v_m\\)\uc5d0 \ub300\ud574<br \/>\n\\[<br \/>\nc_1v_1+\\cdots+c_mv_m=0<br \/>\n\\]<br \/>\n\uc774\uba74 \ubc18\ub4dc\uc2dc \\(c_1=\\cdots=c_m=0\\)\uc77c \ub54c \\(S\\)\uac00 <span class=\"defined\">\uc77c\ucc28\ub3c5\ub9bd<\/span>(linearly independent)\uc774\ub77c\uace0 \ud55c\ub2e4. \\(S\\)\uac00 \\(V\\)\ub97c \uc0dd\uc131\ud558\uba74\uc11c \uc77c\ucc28\ub3c5\ub9bd\uc774\uba74 \\(S\\)\ub97c \\(V\\)\uc758 <span class=\"defined\">\uae30\uc800<\/span>(basis), \ubb34\ud55c\uae09\uc218\ub85c \uc815\uc758\ub418\ub294 \ub2e4\ub978 \uc885\ub958\uc758 \uae30\uc800\uc640 \uad6c\ubcc4\ud560 \ub54c\ub294 <span class=\"defined\">\ud558\uba5c \uae30\uc800<\/span>(Hamel basis)\ub77c\uace0 \ud55c\ub2e4. \uc720\ud55c\ud55c \uae30\uc800\ub97c \uac16\ub294 \uacf5\uac04\uc744 <span class=\"defined\">\uc720\ud55c\ucc28\uc6d0<\/span>(finite-dimensional) \ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud55c\ub2e4.<\/p>\n<p>\uae30\uc800\ub97c \uad6c\uc131\ud558\ub294 \uc6d0\uc18c\uc758 \uac1c\uc218\ub97c \ucc28\uc6d0\uc774\ub77c \ubd80\ub974\ub824\uba74, \ud55c \ubca1\ud130\uacf5\uac04\uc758 \uc11c\ub85c \ub2e4\ub978 \ub450 \uc720\ud55c \uae30\uc800\uac00 \ud56d\uc0c1 \uac19\uc740 \uc218\uc758 \uc6d0\uc18c\ub97c \uac16\ub294\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uba3c\uc800 \uc99d\uba85\ud574\uc57c \ud55c\ub2e4. \uc774\ub97c \ubcf4\uc7a5\ud558\ub294 \uacb0\uacfc\uac00 \ub2e4\uc74c \uad50\ud658 \ubcf4\uc870\uc815\ub9ac\uc774\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p>\n<span class=\"theorem\">\uc815\ub9ac: \uc288\ud0c0\uc774\ub2c8\uce20 \uad50\ud658 \ubcf4\uc870\uc815\ub9ac (\uc720\ud55c \ub9ac\uc2a4\ud2b8 \ud615\ud0dc)<\/span><br \/>\n\ubca1\ud130\uacf5\uac04 \\(V\\)\uc5d0\uc11c \uc720\ud55c \ub9ac\uc2a4\ud2b8 \\((v_1,\\ldots,v_m)\\)\uc774 \\(V\\)\ub97c \uc0dd\uc131\ud558\uace0 \ub2e4\ub978 \uc720\ud55c \ub9ac\uc2a4\ud2b8 \\((w_1,\\ldots,w_n)\\)\uc774 \uc77c\ucc28\ub3c5\ub9bd\uc774\uba74 \\(n\\le m\\)\uc774\ub2e4. \ub354 \ub098\uc544\uac00 \\(v\\)\ub4e4\uc758 \uc21c\uc11c\ub97c \uc801\uc808\ud788 \ubc14\uafb8\uc5b4<br \/>\n\\[<br \/>\n(w_1,\\ldots,w_n,v_{n+1},\\ldots,v_m)<br \/>\n\\]<br \/>\n\uc774 \\(V\\)\ub97c \uc0dd\uc131\ud558\uac8c \ud560 \uc218 \uc788\ub2e4.\n<\/p>\n<\/div>\n<p>\uc99d\uba85\uc744 \uc704\ud574 \\(0\\le k\\le\\min(m,n)\\)\uc778 \uac01 \\(k\\)\uc5d0 \ub300\ud558\uc5ec, \uc544\uc9c1 \ub0a8\uc544 \uc788\ub294 \\(v\\)\ub4e4\uc758 \uc21c\uc11c\ub97c \ud544\uc694\uc5d0 \ub530\ub77c \ubc14\uafb8\uba74<br \/>\n\\[<br \/>\n(w_1,\\ldots,w_k,v_{k+1},\\ldots,v_m)<br \/>\n\\]<br \/>\n\uc774 \\(V\\)\ub97c \uc0dd\uc131\ud55c\ub2e4\ub294 \uc8fc\uc7a5\uc744 \uadc0\ub0a9\ubc95\uc73c\ub85c \ubcf4\uc774\uc790.<\/p>\n<p>\\(k=0\\)\uc77c \ub54c\ub294 \\((v_1,\\ldots,v_m)\\)\uc774 \\(V\\)\ub97c \uc0dd\uc131\ud55c\ub2e4\ub294 \uac00\uc815\uacfc \uac19\ub2e4. \uc774\uc81c \\(k-1\\)\uc77c \ub54c<br \/>\n\\[<br \/>\n(w_1,\\ldots,w_{k-1},v_k,\\ldots,v_m)<br \/>\n\\]<br \/>\n\uc774 \\(V\\)\ub97c \uc0dd\uc131\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \\(w_k\\)\ub3c4 \uc774 \ub9ac\uc2a4\ud2b8\uc758 \uc77c\ucc28\uacb0\ud569\uc774\ubbc0\ub85c<br \/>\n\\[<br \/>\nw_k=c_1w_1+\\cdots+c_{k-1}w_{k-1}+d_kv_k+\\cdots+d_mv_m<br \/>\n\\]<br \/>\n\uc73c\ub85c \uc4f8 \uc218 \uc788\ub2e4. \uacc4\uc218 \\(d_k,\\ldots,d_m\\) \uac00\uc6b4\ub370 \uc801\uc5b4\ub3c4 \ud558\ub098\ub294 \\(0\\)\uc774 \uc544\ub2c8\ub2e4. \ubaa8\ub450 \\(0\\)\uc774\ub77c\uba74 \\(w_k\\)\uac00 \\(w_1,\\ldots,w_{k-1}\\)\uc758 \uc77c\ucc28\uacb0\ud569\uc774 \ub418\uc5b4 \\((w_1,\\ldots,w_n)\\)\uc758 \uc77c\ucc28\ub3c5\ub9bd\uc131\uacfc \ubaa8\uc21c\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<p>\\(d_j\\ne0\\)\uc778 \uc5b4\ub5a4 \\(j\\ge k\\)\ub97c \uace0\ub978 \ub4a4, \uc544\uc9c1 \ub0a8\uc544 \uc788\ub294 \\(v_j\\)\uc640 \\(v_k\\)\uc758 \uc774\ub984\ub9cc \ub9de\ubc14\uafb8\uba74 \\(d_k\\ne0\\)\uc774\ub77c\uace0 \ub458 \uc218 \uc788\ub2e4. \uc704 \ub4f1\uc2dd\uc744 \\(v_k\\)\uc5d0 \ub300\ud558\uc5ec \ud480\uba74<br \/>\n\\[<br \/>\nv_k=\\frac{1}{d_k}\\left(w_k-c_1w_1-\\cdots-c_{k-1}w_{k-1}-d_{k+1}v_{k+1}-\\cdots-d_mv_m\\right)<br \/>\n\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \ub530\ub77c\uc11c \uae30\uc874 \uc0dd\uc131 \ub9ac\uc2a4\ud2b8\uc5d0\uc11c \\(v_k\\)\ub97c \\(w_k\\)\ub85c \ubc14\uafb8\uc5b4\ub3c4 \\(V\\) \uc804\uccb4\ub97c \uc0dd\uc131\ud55c\ub2e4. \uadc0\ub0a9\uc801\uc73c\ub85c<br \/>\n\\[<br \/>\n(w_1,\\ldots,w_k,v_{k+1},\\ldots,v_m)<br \/>\n\\]<br \/>\n\uc774 \\(V\\)\ub97c \uc0dd\uc131\ud55c\ub2e4.<\/p>\n<p>\uc774\uc81c \\(n>m\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uc704 \uc8fc\uc7a5\uc744 \\(k=m\\)\uc5d0 \uc801\uc6a9\ud558\uba74 \\((w_1,\\ldots,w_m)\\)\ub9cc\uc73c\ub85c \\(V\\)\ub97c \uc0dd\uc131\ud55c\ub2e4. \ub530\ub77c\uc11c \\(w_{m+1}\\)\uc740 \uc774\ub4e4\uc758 \uc77c\ucc28\uacb0\ud569\uc778\ub370, \uc774\ub294 \\((w_1,\\ldots,w_n)\\)\uc774 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub77c\ub294 \uac00\uc815\uacfc \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(n\\le m\\)\uc774\ub2e4. \ub610\ud55c \\(n\\le m\\)\uc778 \uacbd\uc6b0 \uadc0\ub0a9 \uc8fc\uc7a5\uc744 \\(k=n\\)\uae4c\uc9c0 \uc801\uc6a9\ud558\uba74 \ubcf4\uc870\uc815\ub9ac\uc758 \ub354 \uac15\ud55c \uc0dd\uc131 \uacb0\ub860\ub3c4 \uc5bb\ub294\ub2e4.<span class=\"qed\"><\/span><\/p>\n<p>\uc774\uc81c \\(V\\)\uc5d0 \ub450 \uae30\uc800 \\((v_1,\\ldots,v_m)\\)\uacfc \\((w_1,\\ldots,w_n)\\)\uc774 \uc788\ub2e4\uace0 \ud558\uc790. \uccab \ubc88\uc9f8 \uae30\uc800\ub294 \uc0dd\uc131 \ub9ac\uc2a4\ud2b8\uc774\uace0 \ub450 \ubc88\uc9f8 \uae30\uc800\ub294 \uc77c\ucc28\ub3c5\ub9bd \ub9ac\uc2a4\ud2b8\uc774\ubbc0\ub85c \\(n\\le m\\)\uc774\ub2e4. \uc5ed\ud560\uc744 \ubc14\uafb8\uba74 \\(m\\le n\\)\ub3c4 \uc131\ub9bd\ud558\ubbc0\ub85c \\(m=n\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ud55c \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc758 \ubaa8\ub4e0 \uae30\uc800\ub294 \uac19\uc740 \uc218\uc758 \uc6d0\uc18c\ub97c \uac16\ub294\ub2e4. \uc774 \uacf5\ud1b5\uac12\uc744 \\(V\\)\uc758 <span class=\"defined\">\ucc28\uc6d0<\/span>(dimension)\uc774\ub77c \ud558\uace0 \\(\\dim V\\)\ub85c \uc4f4\ub2e4.<\/p>\n<p>\ub450 \ubca1\ud130\uacf5\uac04 \\(V,W\\) \uc0ac\uc774\uc758 \ud568\uc218 \\(\\varphi:V\\to W\\)\uac00 \uc784\uc758\uc758 \\(v,v&#8217;\\in V\\)\uc640 \\(\\alpha\\in F\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\n\\varphi(v+v&#8217;)=\\varphi(v)+\\varphi(v&#8217;),\\quad \\varphi(\\alpha v)=\\alpha\\varphi(v)<br \/>\n\\]<br \/>\n\ub97c \ub9cc\uc871\ud558\uba74 \\(\\varphi\\)\ub97c <span class=\"defined\">\uc120\ud615\uc0ac\uc0c1<\/span>(linear map)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc804\ub2e8\uc0ac\uc778 \uc120\ud615\uc0ac\uc0c1\uc744 <span class=\"defined\">\ub3d9\ud615\uc0ac\uc0c1<\/span>(isomorphism)\uc774\ub77c \ud558\uace0, \uadf8\ub7ec\ud55c \uc0ac\uc0c1\uc774 \uc874\uc7ac\ud558\ub294 \ub450 \ubca1\ud130\uacf5\uac04\uc744 \uc11c\ub85c \ub3d9\ud615\uc774\ub77c\uace0 \ud55c\ub2e4.<\/p>\n<div class=\"box theorem\">\n<p>\n<span class=\"theorem\">\uc815\ub9ac: \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc758 \ubd84\ub958<\/span><br \/>\n\uac19\uc740 \uccb4 \\(F\\) \uc704\uc758 \ub450 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04 \\(V,W\\)\uac00 \uc11c\ub85c \ub3d9\ud615\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740<br \/>\n\\[<br \/>\n\\dim V=\\dim W<br \/>\n\\]<br \/>\n\uc778 \uac83\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uba3c\uc800 \\(\\dim V=\\dim W=n\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uace0 \uac01\uac01\uc758 \uae30\uc800 \\((v_1,\\ldots,v_n)\\), \\((w_1,\\ldots,w_n)\\)\uc744 \uc7a1\uc790. \uae30\uc800\uc758 \uc0dd\uc131\uc131\uacfc \uc77c\ucc28\ub3c5\ub9bd\uc131 \ub54c\ubb38\uc5d0 \\(V\\)\uc758 \ubaa8\ub4e0 \ubca1\ud130\ub294<br \/>\n\\[<br \/>\nv=c_1v_1+\\cdots+c_nv_n<br \/>\n\\]<br \/>\n\uc73c\ub85c \uc720\uc77c\ud558\uac8c \ud45c\ud604\ub41c\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[<br \/>\n\\varphi(c_1v_1+\\cdots+c_nv_n)=c_1w_1+\\cdots+c_nw_n<br \/>\n\\]<br \/>\n\uc73c\ub85c \\(\\varphi:V\\to W\\)\ub97c \uc798 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc88c\ud45c\uc758 \ub367\uc148\uacfc \uc2a4\uce7c\ub77c\ubc30\uac00 \uccb4 \\(F\\)\uc758 \uc5f0\uc0b0\uc5d0 \ub530\ub77c \uc131\ubd84\ubcc4\ub85c \uc774\ub8e8\uc5b4\uc9c0\ubbc0\ub85c \\(\\varphi\\)\ub294 \uc120\ud615\uc774\ub2e4. \\((w_1,\\ldots,w_n)\\)\uc774 \\(W\\)\ub97c \uc0dd\uc131\ud558\ubbc0\ub85c \\(\\varphi\\)\ub294 \uc804\uc0ac\uc774\uace0, \uc774 \ub9ac\uc2a4\ud2b8\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc774\ubbc0\ub85c \\(\\varphi(v)=0\\)\uc774\uba74 \\(v=0\\)\uc774\ub2e4.<\/p>\n<p>[\uc120\ud615\uc0ac\uc0c1 \\(T:V\\to W\\)\ub294 \\(\\ker T=\\{0\\}\\)\uc77c \ub54c \uadf8\ub9ac\uace0 \uadf8\ub54c\uc5d0\ub9cc \ub2e8\uc0ac\uc774\ub2e4. \uc2e4\uc81c\ub85c \\(T(v)=T(w)\\)\uc774\uba74 \\(T(v-w)=0\\)\uc774\ubbc0\ub85c \\(\\ker T=\\{0\\}\\)\uc77c \ub54c \\(v=w\\)\uc774\ub2e4. \ubc18\ub300\ub85c \\(T\\)\uac00 \ub2e8\uc0ac\uc774\uba74 \\(T(v)=0=T(0)\\)\uc5d0\uc11c \\(v=0\\)\uc774\ub2e4.]<\/p>\n<p>\ub530\ub77c\uc11c \\(\\varphi\\)\ub294 \uc804\ub2e8\uc0ac \uc120\ud615\uc0ac\uc0c1\uc774\uace0 \\(V\\)\uc640 \\(W\\)\ub294 \ub3d9\ud615\uc774\ub2e4. \uc5ed\uc73c\ub85c \\(V\\)\uc640 \\(W\\) \uc0ac\uc774\uc5d0 \ub3d9\ud615\uc0ac\uc0c1 \\(\\varphi:V\\to W\\)\uac00 \uc788\uace0 \\((v_1,\\ldots,v_n)\\)\uc774 \\(V\\)\uc758 \uae30\uc800\ub77c\uace0 \ud558\uc790. \\(\\varphi\\)\uc758 \uc804\uc0ac\uc131 \ub54c\ubb38\uc5d0 \\((\\varphi(v_1),\\ldots,\\varphi(v_n))\\)\uc740 \\(W\\)\ub97c \uc0dd\uc131\ud55c\ub2e4. \ub610\ud55c<br \/>\n\\[<br \/>\nc_1\\varphi(v_1)+\\cdots+c_n\\varphi(v_n)=0<br \/>\n\\]<br \/>\n\uc774\uba74 \uc120\ud615\uc131\uc73c\ub85c \\(\\varphi(c_1v_1+\\cdots+c_nv_n)=0\\)\uc774\uace0, \\(\\varphi\\)\uc758 \ub2e8\uc0ac\uc131\uacfc \\((v_1,\\ldots,v_n)\\)\uc758 \uc77c\ucc28\ub3c5\ub9bd\uc131 \ub54c\ubb38\uc5d0 \ubaa8\ub4e0 \\(c_i=0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\((\\varphi(v_1),\\ldots,\\varphi(v_n))\\)\uc740 \\(W\\)\uc758 \uae30\uc800\uc774\uace0 \\(\\dim W=n=\\dim V\\)\uc774\ub2e4.<span class=\"qed\"><\/span><\/p>\n<p>\uc774 \uc815\ub9ac\ub294 \uac19\uc740 \uccb4 \uc704\uc758 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc744 <em>\ubca1\ud130\uacf5\uac04\uc73c\ub85c\uc11c<\/em> \ubd84\ub958\ud558\ub294 \uc815\ubcf4\uac00 \ucc28\uc6d0 \ud558\ub098\ubfd0\uc784\uc744 \ub9d0\ud55c\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(P_n(\\mathbb R)\\)\uc758 \ud45c\uc900 \uae30\uc800\ub294 \\((1,x,\\ldots,x^n)\\)\uc774\ubbc0\ub85c \ucc28\uc6d0\uc774 \\(n+1\\)\uc774\uace0,<br \/>\n\\[<br \/>\na_0+a_1x+\\cdots+a_nx^n\\longmapsto(a_0,a_1,\\ldots,a_n)<br \/>\n\\]<br \/>\n\uc740 \\(P_n(\\mathbb R)\\)\uacfc \\(\\mathbb R^{n+1}\\) \uc0ac\uc774\uc758 \uba85\uc2dc\uc801\uc778 \uc120\ud615\ub3d9\ud615\uc0ac\uc0c1\uc774\ub2e4. \uc55e\uc758 \ubbf8\ubd84\ubc29\uc815\uc2dd \ud574\uacf5\uac04\ub3c4 \\((\\cos x,\\sin x)\\)\ub97c \uae30\uc800\ub85c \uac00\uc9c0\ubbc0\ub85c \\(\\mathbb R^2\\)\uc640 \ub3d9\ud615\uc774\ub2e4. <!-- \ub2e4\ub9cc \uc774 \uacb0\ub860\uc740 \ub2e4\ud56d\uc2dd\uc758 \uacf1\uc148, \ud2b9\uc815\ud55c \ub178\ub984\uc774\ub098 \ub0b4\uc801 \uac19\uc740 \ucd94\uac00 \uad6c\uc870\uae4c\uc9c0 \ubcf4\uc874\ub41c\ub2e4\ub294 \ub73b\uc740 \uc544\ub2c8\ub2e4. --> \ucd94\uc0c1\ud654\ub294 \uc5b4\ub5a4 \uad6c\uc870\ub97c \ubcf4\uc874\ud558\uace0 \uc5b4\ub5a4 \uad6c\uc870\ub97c \uc78a\uc5c8\ub294\uc9c0 \ubd84\uba85\ud788 \ud560 \ub54c \uc815\ud655\ud55c \ud798\uc744 \ubc1c\ud718\ud55c\ub2e4.<\/p>\n<h4>\ubb34\ud55c\ucc28\uc6d0\uc73c\ub85c \uac00\ub294 \uc0ac\ub2e4\ub9ac<\/h4>\n<p>\ubb34\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc740 \uc720\ud55c\ud55c \ud558\uba5c \uae30\uc800\ub97c \uac16\uc9c0 \uc54a\ub294 \uacf5\uac04\uc774\ub2e4. \ud558\uba5c \uae30\uc800\uc5d0\uc11c\ub294 \uac01 \ubca1\ud130\uac00 \uc5b8\uc81c\ub098 <em>\uc720\ud55c<\/em> \uc77c\ucc28\uacb0\ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4. \uadf8\ub7ec\ub098 \ud568\uc218\ud574\uc11d\ud559\uc5d0\uc11c \uc911\uc694\ud55c \ud478\ub9ac\uc5d0 \uae09\uc218, \ud568\uc218\uc5f4\uc758 \uadf9\ud55c, \ubb34\ud55c\ud569\uc744 \ub2e4\ub8e8\ub824\uba74 \ubca1\ud130\uacf5\uac04\uc758 \ub300\uc218 \uacf5\ub9ac\ub9cc\uc73c\ub85c\ub294 \ubd80\uc871\ud558\ub2e4. \uc5b4\ub5a4 \uc218\uc5f4\uc774 \uc11c\ub85c \uac00\uae4c\uc6cc\uc9c0\ub294\uc9c0, \uadf8 \uadf9\ud55c\uc774 \uacf5\uac04 \uc548\uc5d0 \uba38\ubb34\ub294\uc9c0\ub97c \ub9d0\ud574 \uc8fc\ub294 \ub178\ub984\uc774\ub098 \uc704\uc0c1 \uac19\uc740 \ucd94\uac00 \uad6c\uc870\uac00 \ud544\uc694\ud558\ub2e4.<a class=\"math-abs-series-ref\" title=\"Hunter, J. K., &amp; Nachtergaele, B. (2001). Applied Analysis. World Scientific.\" href=\"#ref-HunterNachtergaele2001\">[HunterNachtergaele2001]<\/a><\/p>\n<p>\uccb4 \\(F=\\mathbb R\\) \ub610\ub294 \\(\\mathbb C\\) \uc704\uc758 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc5d0\uc11c <span class=\"defined\">\ub178\ub984<\/span>(norm)\uc740 \ud568\uc218 \\(\\|\\cdot\\|:V\\to[0,\\infty)\\)\ub85c\uc11c \ub2e4\uc74c\uc744 \ub9cc\uc871\ud55c\ub2e4.<br \/>\n\\[<br \/>\n\\|v\\|=0\\iff v=0,\\quad \\|\\alpha v\\|=|\\alpha|\\,\\|v\\|,\\quad \\|v+w\\|\\le\\|v\\|+\\|w\\|.<br \/>\n\\]<br \/>\n\ub178\ub984\uc740 \\(d(v,w)=\\|v-w\\|\\)\ub77c\ub294 \uac70\ub9ac\ub97c \ub9cc\ub4e4\uba70, \uc774 \uac70\ub9ac\uc5d0\uc11c \uc218\uc5f4\uc758 \uc218\ub834\uacfc \ucf54\uc2dc\uc218\uc5f4\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \ubaa8\ub4e0 \ucf54\uc2dc\uc218\uc5f4\uc774 \\(V\\) \uc548\uc758 \uc5b4\ub5a4 \uc6d0\uc18c\ub85c \uc218\ub834\ud558\uba74 \\(V\\)\uac00 \uadf8 \ub178\ub984\uc5d0 \uad00\ud558\uc5ec <span class=\"defined\">\uc644\ube44<\/span>(complete)\ub77c\uace0 \ud55c\ub2e4. \uc644\ube44 \ub178\ub984\ubca1\ud130\uacf5\uac04\uc774 <span class=\"defined\">\ubc14\ub098\ud750\uacf5\uac04<\/span>(Banach space)\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Hunter, J. K., &amp; Nachtergaele, B. (2001). Applied Analysis. World Scientific.\" href=\"#ref-HunterNachtergaele2001\">[HunterNachtergaele2001]<\/a><\/p>\n<p>[\uc644\ube44\uc131\uc740 \uc120\ud0dd\ud55c \ub178\ub984\uc5d0 \uad00\ud55c \uc815\ud655\ud55c \uc870\uac74\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\([0,1]\\) \uc704\uc758 \ub2e4\ud56d\uc2dd \uacf5\uac04\uc5d0 \ucd5c\ub313\uac12 \ub178\ub984 \\(\\|p\\|_\\infty=\\max_{x\\in[0,1]}|p(x)|\\)\uc744 \uc8fc\uc790. \ub2e4\ud56d\uc2dd<br \/>\n\\[<br \/>\np_N(x)=\\sum_{k=0}^{N}\\frac{x^k}{k!}<br \/>\n\\]<br \/>\n\uc740 \uc774 \ub178\ub984\uc5d0\uc11c \ucf54\uc2dc\uc218\uc5f4\uc774\uc9c0\ub9cc \\(C([0,1])\\)\uc5d0\uc11c\uc758 \uadf9\ud55c \\(e^x\\)\ub294 \ub2e4\ud56d\uc2dd\uc774 \uc544\ub2c8\ub2e4. \ub530\ub77c\uc11c \ub2e4\ud56d\uc2dd \uacf5\uac04 \uc548\uc5d0\ub294 \uc774 \uc218\uc5f4\uc758 \uadf9\ud55c\uc774 \uc5c6\uace0, \uadf8 \uacf5\uac04\uc740 \uc644\ube44\uac00 \uc544\ub2c8\ub2e4.<a class=\"math-abs-series-ref\" title=\"Hunter, J. K., &amp; Nachtergaele, B. (2001). Applied Analysis. World Scientific.\" href=\"#ref-HunterNachtergaele2001\">[HunterNachtergaele2001]<\/a>]<\/p>\n<p>\uc77c\ubc18\uc801\uc778 \ub178\ub984\ub9cc\uc73c\ub85c\ub294 \ub450 \ubca1\ud130 \uc0ac\uc774\uc758 \uc9c1\uad50\ub97c \uc815\ud560 \uc218 \uc5c6\ub2e4. \uc774\ub97c \uc704\ud574 \\(F=\\mathbb R\\) \ub610\ub294 \\(\\mathbb C\\) \uc704\uc758 \ubca1\ud130\uacf5\uac04 \\(V\\)\uc5d0 <span class=\"defined\">\ub0b4\uc801<\/span>(inner product) \\(\\langle\\cdot,\\cdot\\rangle:V\\times V\\to F\\)\uc744 \ub454\ub2e4. \ub0b4\uc801\uc740 \uc591\uc758 \uc815\ubd80\ud638\uc774\uace0 \ucf24\ub808\ub300\uce6d\uc774\uba70, \ud55c \ubcc0\uc218\uc5d0\uc11c\ub294 \uc120\ud615\uc774\uace0 \ub2e4\ub978 \ubcc0\uc218\uc5d0\uc11c\ub294 \ucf24\ub808\uc120\ud615\uc774\ub2e4. \uc2e4\uc218 \ubca1\ud130\uacf5\uac04\uc5d0\uc11c\ub294 \ub300\uce6d\uc778 \uc30d\uc120\ud615 \ud615\uc2dd\uc774 \ub41c\ub2e4. \ub0b4\uc801\uc740<br \/>\n\\[<br \/>\n\\|v\\|=\\sqrt{\\langle v,v\\rangle}<br \/>\n\\]<br \/>\n\ub85c \ub178\ub984\uc744 \ub9cc\ub4e4\uace0, \\(\\langle v,w\\rangle=0\\)\uc774\uba74 \\(v\\)\uc640 \\(w\\)\uac00 <span class=\"defined\">\uc9c1\uad50<\/span>(orthogonal)\ud55c\ub2e4\uace0 \ud55c\ub2e4. \uc774 \uc720\ub3c4\ub41c \ub178\ub984\uc5d0 \uad00\ud558\uc5ec \uc644\ube44\uc778 \ub0b4\uc801\uacf5\uac04\uc744 <span class=\"defined\">\ud790\ubca0\ub974\ud2b8\uacf5\uac04<\/span>(Hilbert space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<a class=\"math-abs-series-ref\" title=\"Hunter, J. K., &amp; Nachtergaele, B. (2001). Applied Analysis. World Scientific.\" href=\"#ref-HunterNachtergaele2001\">[HunterNachtergaele2001]<\/a><\/p>\n<p>[\ud790\ubca0\ub974\ud2b8, \uc5d0\ub974\ud558\ub974\ud2b8 \uc288\ubbf8\ud2b8, \ud504\ub9ac\uc81c\uc2dc \ub9ac\uc2a4 \ub4f1\uc740 20\uc138\uae30 \ucd08 \uc801\ubd84\ubc29\uc815\uc2dd\uacfc \uad6c\uccb4\uc801\uc778 \uc218\uc5f4\u00b7\ud568\uc218\uacf5\uac04\uc744 \uc5f0\uad6c\ud558\uba70 \uc774 \uc774\ub860\uc758 \ud1a0\ub300\ub97c \ub9cc\ub4e4\uc5c8\ub2e4. \ud3f0 \ub178\uc774\ub9cc(John von Neumann)\uc740 1927\ub144 \ubb34\ub835 \uc774 \ubaa8\ub378\ub4e4\uc744 \ud604\ub300\uc801\uc778 \ucd94\uc0c1 \ubcf5\uc18c \ud790\ubca0\ub974\ud2b8\uacf5\uac04\uc758 \ud2c0\ub85c \ud1b5\ud569\ud558\uace0, \uc774\ub97c \uc591\uc790\uc5ed\ud559\uc758 \uc5c4\ubc00\ud55c \uc218\ud559\uc801 \ud615\uc2dd\uc5d0 \uc801\uc6a9\ud588\ub2e4. <!-- \ub530\ub77c\uc11c \ud3f0 \ub178\uc774\ub9cc\uc774 \uc120\ud589 \uc5f0\uad6c \uc5c6\uc774 \uc774 \uacf5\uac04\uc744 \ucc98\uc74c \ub9cc\ub4e4\uc5c8\ub2e4\uace0 \ud558\uae30\ubcf4\ub2e4 \uae30\uc874 \ubd84\uc11d\uc758 \uc5ec\ub7ec \ubaa8\ub378\uc744 \ud558\ub098\uc758 \ucd94\uc0c1 \uad6c\uc870\ub85c \uacb0\ud569\ud588\ub2e4\uace0 \ud45c\ud604\ud558\ub294 \ud3b8\uc774 \uc815\ud655\ud558\ub2e4.<a class=\"math-abs-series-ref\" title=\"von Neumann, J. (1927). \u201cMathematische Begr\u00fcndung der Quantenmechanik.\u201d Nachrichten von der Gesellschaft der Wissenschaften zu G\u00f6ttingen, Mathematisch-Physikalische Klasse, 1\u201357. EuDML \uc6d0\ubb38.\" href=\"#ref-VonNeumann1927\">[VonNeumann1927]<\/a><a class=\"math-abs-series-ref\" title=\"Landsman, K. (2019). \u201cQuantum Theory and Functional Analysis.\u201d arXiv:1911.06630.\" href=\"#ref-Landsman2019\">[Landsman2019]<\/a>] --><\/p>\n<p>\uc774 \ucd94\uc0c1\ud654\uac00 \uc5b4\ub5bb\uac8c \uc720\ud55c\ucc28\uc6d0 \uae30\ud558\ud559\uc744 \ud568\uc218\uc758 \uc138\uacc4\ub85c \uc62e\uae30\ub294\uc9c0\ub294 \ud478\ub9ac\uc5d0 \uae09\uc218\uc5d0\uc11c \uc798 \ub4dc\ub7ec\ub09c\ub2e4. \uc2e4\uc218 \ud790\ubca0\ub974\ud2b8\uacf5\uac04<br \/>\n\\[<br \/>\nH=L^2([-\\pi,\\pi];\\mathbb R),\\quad<br \/>\n\\langle f,g\\rangle=\\int_{-\\pi}^{\\pi}f(x)g(x)\\,dx<br \/>\n\\]<br \/>\n\ub97c \uc0dd\uac01\ud558\uc790. \uc5ec\uae30\uc11c\ub294 \uac70\uc758 \ubaa8\ub4e0 \uacf3\uc5d0\uc11c \uac12\uc774 \uac19\uc740 \ud568\uc218\ub4e4\uc744 \uac19\uc740 \uc6d0\uc18c\ub85c \ubcf8\ub2e4. \ud568\uc218\ub4e4<br \/>\n\\[<br \/>\ne_0(x)=\\frac{1}{\\sqrt{2\\pi}},\\quad<br \/>\ne_n^c(x)=\\frac{\\cos(nx)}{\\sqrt{\\pi}},\\quad<br \/>\ne_n^s(x)=\\frac{\\sin(nx)}{\\sqrt{\\pi}}\\quad(n\\ge1)<br \/>\n\\]<br \/>\n\uc740 \\(H\\)\uc758 \uc815\uaddc\uc9c1\uad50\uae30\uc800\ub97c \uc774\ub8ec\ub2e4. \uc5ec\uae30\uc11c \u2018\uae30\uc800\u2019\ub294 \uc55e \uc808\uc758 \ud558\uba5c \uae30\uc800\uc640 \ub73b\uc774 \ub2e4\ub974\ub2e4. \uc774 \ud568\uc218\ub4e4\uc758 \uc720\ud55c \uc77c\ucc28\uacb0\ud569\ub9cc\uc73c\ub85c \\(H\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ub97c \uc815\ud655\ud788 \uc5bb\ub294\ub2e4\ub294 \ub73b\uc774 \uc544\ub2c8\ub77c, \uadf8 \uc720\ud55c \uc120\ud615\uc0dd\uc131\uacf5\uac04\uc774 \\(H\\)\uc5d0\uc11c \uc870\ubc00\ud558\ub2e4\ub294 \ub73b\uc774\ub2e4.<a class=\"math-abs-series-ref\" title=\"Hunter, J. K., &amp; Nachtergaele, B. (2001). Applied Analysis. World Scientific.\" href=\"#ref-HunterNachtergaele2001\">[HunterNachtergaele2001]<\/a><\/p>\n<p>\ub530\ub77c\uc11c \ubaa8\ub4e0 \\(f\\in H\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nS_Nf=\\langle f,e_0\\rangle e_0+<br \/>\n\\sum_{n=1}^{N}\\left(\\langle f,e_n^c\\rangle e_n^c+\\langle f,e_n^s\\rangle e_n^s\\right)<br \/>\n\\]<br \/>\n\ub294 \uc720\ud55c\ucc28\uc6d0 \uc0bc\uac01\ub2e4\ud56d\uc2dd \uacf5\uac04\uc73c\ub85c\uc758 \uc9c1\uad50\uc0ac\uc601\uc774\uace0,<br \/>\n\\[<br \/>\n\\|f-S_Nf\\|_2\\longrightarrow0<br \/>\n\\]<br \/>\n\uc774\ub2e4. <!-- \uc774\ub294 \\(L^2\\) \ub178\ub984\uc5d0\uc11c\uc758 \ud3c9\uade0\uc81c\uacf1 \uc218\ub834\uc774\uc9c0, \ubaa8\ub4e0 \uc810\uc5d0\uc11c\uc758 \uc810\ubcc4\uc218\ub834\uc744 \uc790\ub3d9\uc73c\ub85c \ub73b\ud558\uc9c0 \uc54a\ub294\ub2e4. --> \ub610\ud55c<br \/>\n\\[<br \/>\n\\|f\\|_2^2=<br \/>\n|\\langle f,e_0\\rangle|^2+<br \/>\n\\sum_{n=1}^{\\infty}\\left(<br \/>\n|\\langle f,e_n^c\\rangle|^2+<br \/>\n|\\langle f,e_n^s\\rangle|^2<br \/>\n\\right)<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\uac83\uc774 \uc774 \uacbd\uc6b0\uc758 <span class=\"defined\">\ud30c\ub974\uc138\ubc1c \ub4f1\uc2dd<\/span>(Parseval&#8217;s identity)\uc774\uba70, \uc815\uaddc\uc9c1\uad50\uc88c\ud45c\uc5d0 \ub300\ud55c \ud53c\ud0c0\uace0\ub77c\uc2a4 \uc815\ub9ac\uc758 \ubb34\ud55c\ucc28\uc6d0 \ud615\ud0dc\uc774\ub2e4. \uc2e0\ud638 \ud574\uc11d\uc5d0\uc11c\ub294 \\(\\|f\\|_2^2\\)\uc744 \uc5d0\ub108\uc9c0\ub85c \ud574\uc11d\ud558\uae30\ub3c4 \ud55c\ub2e4.<a class=\"math-abs-series-ref\" title=\"Hunter, J. K., &amp; Nachtergaele, B. (2001). Applied Analysis. World Scientific.\" href=\"#ref-HunterNachtergaele2001\">[HunterNachtergaele2001]<\/a><\/p>\n<h4>\uc88c\ud45c\ub97c \ubc84\ub9ac\uace0 \uacf5\ub9ac\uac00 \ub0a8\ub2e4<\/h4>\n<p>\ubca1\ud130\uacf5\uac04\uc758 \uc5ed\uc0ac\ub294 \ud55c \ubc88\uc758 \ubc1c\uba85\uc73c\ub85c \ud769\uc5b4\uc9c4 \ubaa8\ub4e0 \uc0ac\ub840\uac00 \uc989\uc2dc \ud1b5\ud569\ub41c \uc774\uc57c\uae30\uac00 \uc544\ub2c8\ub2e4. \uc5f0\ub9bd\ubc29\uc815\uc2dd\uacfc \uae30\ud558\ud559, \ud568\uc218\uc640 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc5d0\uc11c \ubc1c\uc804\ud55c \uc5ec\ub7ec \uc120\ud615\uc801 \ubc29\ubc95\uc774 \uadf8\ub77c\uc2a4\ub9cc\uacfc \ud398\uc544\ub178\uc758 \uc774\ub978 \uc77c\ubc18\ud654\ub97c \uac70\uce58\uace0, 20\uc138\uae30 \ucd08 \ub300\uc218\ud559\uacfc \ud568\uc218\ud574\uc11d\ud559\uc758 \ud544\uc694 \uc18d\uc5d0\uc11c \uacf5\ud1b5\ub41c \uc5b8\uc5b4\ub85c \uc815\ucc29\ud55c \uacfc\uc815\uc774\ub2e4. \uc774 \uc5b8\uc5b4 \uc548\uc5d0\uc11c \uac19\uc740 \uccb4 \uc704\uc758 \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc740 \ucc28\uc6d0\uc73c\ub85c \ubd84\ub958\ub418\uace0, \ub2e4\ud56d\uc2dd\uc774\ub098 \ud568\uc218\uc758 \ud574\uacf5\uac04\uc5d0\ub3c4 \uae30\uc800\uc640 \uc88c\ud45c\ub77c\ub294 \ub3d9\uc77c\ud55c \ub17c\ub9ac\ub97c \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774 \uc5f0\uc7ac\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \uc218\ud559\uc801 \uc870\uc791\uc744 <span class=\"defined\">\uc0ac\ub840\uc758 \ud1b5\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub85c \ud55c\ub2e4. \uc804\ud600 \ub2ec\ub77c \ubcf4\uc774\ub294 \ub300\uc0c1\ub4e4\uc5d0\uc11c \uacf5\ud1b5\ub41c \uc5f0\uc0b0\uacfc \ubc95\uce59\uc744 \ubd84\ub9ac\ud558\uace0, \uadf8 \ubc95\uce59\uc744 \ub9cc\uc871\ud558\ub294 \ubaa8\ub4e0 \uc0ac\ub840\uc5d0 \uc801\uc6a9\ub418\ub294 \uc774\ub860\uc744 \uc138\uc6b0\ub294 \uacfc\uc815\uc774\ub2e4. \ub2e4\ub9cc \ucd94\uc0c1\ud654\uac00 \ub300\uc0c1\uc758 \ubaa8\ub4e0 \uc131\uc9c8\uc744 \ubcf4\uc874\ud558\ub294 \uac83\uc740 \uc544\ub2c8\ub2e4. \ubca1\ud130\uacf5\uac04\uc73c\ub85c \ucd94\uc0c1\ud654\ud560 \ub54c\ub294 \ub367\uc148\uacfc \uc2a4\uce7c\ub77c\ubc30\ub97c \ub0a8\uae30\uc9c0\ub9cc, \ub2e4\ud56d\uc2dd\uc758 \uacf1\uc148\uc774\ub098 \ud568\uc218\uc758 \ub178\ub984 \uac19\uc740 \uc815\ubcf4\ub294 \ubcc4\ub3c4\ub85c \uc9c0\uc815\ud558\uc9c0 \uc54a\ub294 \ud55c \uc78a\ud78c\ub2e4. \uc5b4\ub5a4 \uad6c\uc870\ub97c \ub0a8\uae30\uace0 \uc5b4\ub5a4 \uad6c\uc870\ub97c \ubc84\ub838\ub294\uc9c0 \ubc1d\ud788\ub294 \uc77c\uc774 \ucd94\uc0c1\ud654\uc758 \uc815\ud655\uc131\uc744 \uacb0\uc815\ud55c\ub2e4.<\/p>\n<p>\ubb34\ud55c\ucc28\uc6d0 \ud574\uc11d\uc73c\ub85c \ub118\uc5b4\uac00\uba74 \ub300\uc218\uc801 \ubca1\ud130\uacf5\uac04 \uad6c\uc870\ub9cc\uc73c\ub85c\ub294 \uadf9\ud55c\uacfc \ubb34\ud55c\uae09\uc218\ub97c \ub2e4\ub8f0 \uc218 \uc5c6\uc73c\ubbc0\ub85c \ub178\ub984, \uc644\ube44\uc131, \ub0b4\uc801\uc744 \ucd94\uac00\ud574\uc57c \ud588\ub2e4. \uc5ec\uae30\uc11c \ucd94\uc0c1\ud654\ub294 \uc88c\ud45c\ub97c \ub2e8\uc21c\ud788 \uc9c0\uc6b0\ub294 \uc791\uc5c5\uc774 \uc544\ub2c8\ub77c, \uc0c8\ub85c\uc6b4 \ubb38\uc81c\uc5d0 \ud544\uc694\ud55c \uad6c\uc870\ub97c \uc815\ud655\ud788 \uace8\ub77c \uacf5\ub9ac\ub85c \ub354\ud558\ub294 \uc791\uc5c5\uc774\uae30\ub3c4 \ud558\ub2e4.<\/p>\n<p><!-- \uc218\uc5f4\uc774 \uc5b4\ub5a4 \uc810\uc5d0 \uac00\uae4c\uc6cc\uc9c0\uace0 \uadf9\ud55c\uc774 \uc874\uc7ac\ud55c\ub2e4\ub294 \uac1c\ub150\uc758 \uc5c4\ubc00\ud654 \uc5ed\uc2dc \ud55c \uc0ac\ub78c\uc758 \ub2e8\ubc88\uc5d0 \uc774\ub8e8\uc5b4\uc9c4 \ubc1c\uba85\uc740 \uc544\ub2c8\uc5c8\ub2e4. --> \uadf9\ud55c \uac1c\ub150\uc774 \uc624\ub298\ub0a0\ucc98\ub7fc \uc815\uad50\ud574\uc9c0\uae30\uae4c\uc9c0\ub294 \uc624\ub79c \uc2dc\uac04 \uc5ec\ub7ec \uc218\ud559\uc790\uc758 \uc190\uae38\uc774 \ud544\uc694\ud588\ub2e4. \ubd80\ub4f1\uc2dd\uc744 \uc774\uc6a9\ud55c \ud574\uc11d\ud559\uc758 \uc5c4\ubc00\ud654\ub294 \ucf54\uc2dc, \ub9ac\ub9cc, \ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4(Karl Weierstrass) \ub4f1\uc744 \uac70\uce58\uba70 \ubc1c\uc804\ud588\uace0, \ud2b9\ud788 \ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4\uc758 \uac15\uc758\ub294 \uc624\ub298\ub0a0\uc758 \\(\\varepsilon\\)-\\(\\delta\\) \uc5b8\uc5b4\ub97c \uccb4\uacc4\ud654\ud558\ub294 \ub370 \uc911\uc694\ud55c \uc5ed\ud560\uc744 \ud588\ub2e4. <!-- \uc774\ub97c \ucc98\uc74c\ubd80\ud130 \u2018\ubb3c\ub9ac\uc801 \uac70\ub9ac\u2019\uc5d0\ub9cc \ubb36\uc5ec \uc788\ub358 \ub17c\ubc95\uc774\ub77c\uace0 \ud45c\ud604\ud558\ub294 \uac83\uc740 \ubd80\uc815\ud655\ud558\ub2e4.<a class=\"math-abs-series-ref\" title=\"Grabiner, J. V. (1983). \u201cWho Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus.\u201d The American Mathematical Monthly, 90(3), 185\u2013194.\" href=\"#ref-Grabiner1983\">[Grabiner1983]<\/a> --> \uace0\uc804\uc801\uc778 \uac70\ub9ac \uac1c\ub150\uc744 \uc5b4\ub514\uae4c\uc9c0 \ucd94\uc0c1\ud654\ud560 \uc218 \uc788\ub294\uc9c0, \uadf8\ub9ac\uace0 \uadf9\ud55c\uc5d0 \uad00\ud55c \uc131\uc9c8\uc774 \uc5b4\ub5bb\uac8c \uacf5\uac04\uc744 \uaddc\uc815\ud558\ub294 \uc815\uc758\uac00 \ub418\ub294\uc9c0\ub294 <a href=\"..\/math-abstraction-07-topological-spaces\/\">7\ubd80<\/a>\uc5d0\uc11c \uc774\uc5b4 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h4>\ucc38\uace0\ubb38\ud5cc<\/h4>\n<ul>\n<li id=\"ref-Dorier1995\">[Dorier1995] Dorier, J.-L. (1995). \u201cA General Outline of the Genesis of Vector Space Theory.\u201d <i>Historia Mathematica<\/i>, 22(3), 227\u2013261. <a href=\"https:\/\/doi.org\/10.1006\/hmat.1995.1024\">https:\/\/doi.org\/10.1006\/hmat.1995.1024<\/a>.<\/li>\n<li id=\"ref-Moore1995\">[Moore1995] Moore, G. H. (1995). \u201cThe Axiomatization of Linear Algebra: 1875\u20131940.\u201d <i>Historia Mathematica<\/i>, 22(3), 262\u2013303. <a href=\"https:\/\/doi.org\/10.1006\/hmat.1995.1025\">https:\/\/doi.org\/10.1006\/hmat.1995.1025<\/a>.<\/li>\n<li id=\"ref-Grcar2011\">[Grcar2011] Grcar, J. F. (2011). \u201cHow Ordinary Elimination Became Gaussian Elimination.\u201d <i>Historia Mathematica<\/i>, 38(2), 163\u2013218. <a href=\"https:\/\/doi.org\/10.1016\/j.hm.2010.06.003\">https:\/\/doi.org\/10.1016\/j.hm.2010.06.003<\/a>, <a href=\"https:\/\/arxiv.org\/abs\/0907.2397\">arXiv:0907.2397<\/a>.<\/li>\n<li id=\"ref-Crowe1994\">[Crowe1994] Crowe, M. J. (1994 [1967]). <i>A History of Vector Analysis: The Evolution of the Idea of a Vectorial System<\/i>. Dover Publications. <a href=\"https:\/\/books.google.com\/books?id=iVFAVqA91h4C\">Google Books<\/a>.<\/li>\n<li id=\"ref-Teschl2012\">[Teschl2012] Teschl, G. (2012). <i>Ordinary Differential Equations and Dynamical Systems<\/i>. Graduate Studies in Mathematics 140, American Mathematical Society. <a href=\"https:\/\/doi.org\/10.1090\/gsm\/140\">https:\/\/doi.org\/10.1090\/gsm\/140<\/a>, <a href=\"https:\/\/www.mat.univie.ac.at\/~gerald\/ftp\/book-ode\/ode.pdf\">PDF<\/a>.<\/li>\n<li id=\"ref-Peano1888\">[Peano1888] Peano, G. (1888). <i>Calcolo geometrico secondo l&#8217;Ausdehnungslehre di H. Grassmann, preceduto dalle operazioni della logica deduttiva<\/i>. Torino: Fratelli Bocca. <a href=\"https:\/\/books.google.com\/books?id=5LJi3dxLzuwC\">https:\/\/books.google.com\/books?id=5LJi3dxLzuwC<\/a>.<\/li>\n<li id=\"ref-FearnleySander1979\">[FearnleySander1979] Fearnley-Sander, D. (1979). \u201cHermann Grassmann and the Creation of Linear Algebra.\u201d <i>The American Mathematical Monthly<\/i>, 86(10), 809\u2013817. <a href=\"https:\/\/doi.org\/10.1080\/00029890.1979.11994921\">https:\/\/doi.org\/10.1080\/00029890.1979.11994921<\/a>.<\/li>\n<li id=\"ref-Banach1922\">[Banach1922] Banach, S. (1922). \u201cSur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales.\u201d <i>Fundamenta Mathematicae<\/i>, 3, 133\u2013181. <a href=\"https:\/\/doi.org\/10.4064\/fm-3-1-133-181\">https:\/\/doi.org\/10.4064\/fm-3-1-133-181<\/a>, <a href=\"https:\/\/eudml.org\/doc\/213289\">EuDML<\/a>.<\/li>\n<li id=\"ref-Pietsch2007\">[Pietsch2007] Pietsch, A. (2007). <i>History of Banach Spaces and Linear Operators<\/i>. Birkh\u00e4user. <a href=\"https:\/\/doi.org\/10.1007\/978-0-8176-4596-0\">https:\/\/doi.org\/10.1007\/978-0-8176-4596-0<\/a>.<\/li>\n<li id=\"ref-Axler2024\">[Axler2024] Axler, S. (2024). <i>Linear Algebra Done Right<\/i>, 4th ed. Springer. <a href=\"https:\/\/doi.org\/10.1007\/978-3-031-41026-0\">https:\/\/doi.org\/10.1007\/978-3-031-41026-0<\/a>, <a href=\"https:\/\/linear.axler.net\/LADR4e.pdf\">PDF<\/a>.<\/li>\n<li id=\"ref-HunterNachtergaele2001\">[HunterNachtergaele2001] Hunter, J. K., &amp; Nachtergaele, B. (2001). <i>Applied Analysis<\/i>. World Scientific. <a href=\"https:\/\/doi.org\/10.1142\/4319\">https:\/\/doi.org\/10.1142\/4319<\/a>, <a href=\"https:\/\/www.math.ucdavis.edu\/~bxn\/applied_analysis.pdf\">PDF<\/a>.<\/li>\n<li id=\"ref-VonNeumann1927\">[VonNeumann1927] von Neumann, J. (1927). \u201cMathematische Begr\u00fcndung der Quantenmechanik.\u201d <i>Nachrichten von der Gesellschaft der Wissenschaften zu G\u00f6ttingen, Mathematisch-Physikalische Klasse<\/i>, 1\u201357. <a href=\"https:\/\/eudml.org\/doc\/59215\">https:\/\/eudml.org\/doc\/59215<\/a>.<\/li>\n<li id=\"ref-Landsman2019\">[Landsman2019] Landsman, K. (2019). \u201cQuantum Theory and Functional Analysis.\u201d <a href=\"https:\/\/arxiv.org\/abs\/1911.06630\">arXiv:1911.06630<\/a>.<\/li>\n<li id=\"ref-Grabiner1983\">[Grabiner1983] Grabiner, J. V. (1983). \u201cWho Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus.\u201d <i>The American Mathematical Monthly<\/i>, 90(3), 185\u2013194. <a href=\"https:\/\/doi.org\/10.1080\/00029890.1983.11971185\">https:\/\/doi.org\/10.1080\/00029890.1983.11971185<\/a>.<\/li>\n<\/ul>\n<h4>\uc800\uc791\uad8c<\/h4>\n<p>\uc774\uc2ac\ube44, 2026. designeralice\uff20daum.net.<\/p>\n<div class=\"math-abs-series-contents\">\n<p class=\"math-abs-series-menu-title\"><a href=\"..\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654<\/a><\/p>\n<ol class=\"math-abs-series-menu-list\">\n        <!-- \ud604\uc7ac \ud398\uc774\uc9c0\uc5d0 \ud574\ub2f9\ud558\ub294 li \ud0dc\uadf8\uc5d0 class=\"math-abs-series-current-page\" \uc18d\uc131 \ucd94\uac00 --><\/p>\n<li><a href=\"..\/math-abstraction-01-essence\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \uae30\ubcf8 \uac1c\ub150<\/a><\/li>\n<li><a href=\"..\/math-abstraction-02-implicit-era\/\">\ucd08\uae30 \uc218\ud559\uc5d0\uc11c\uc758 \uc554\ubb35\uc801 \ucd94\uc0c1\ud654<\/a><\/li>\n<li><a href=\"..\/math-abstraction-03-algebraic-symbols\/\">\ub300\uc218\uc801 \uae30\ud638\uc640 \uc5f0\uc0b0 \ubc95\uce59<\/a><\/li>\n<li><a href=\"..\/math-abstraction-04-axiomatic-method\/\">\uacf5\ub9ac\uc801 \ubc29\ubc95\uc5d0 \uc758\ud55c \ucd94\uc0c1\ud654<\/a><\/li>\n<li><a href=\"..\/math-abstraction-05-mathematical-structures\/\">\uc218\ud559\uc801 \uad6c\uc870\uc640 \ubc94\uc8fc\ub860<\/a><\/li>\n<li class=\"math-abs-series-current-page\"><a href=\"..\/math-abstraction-06-vector-spaces\/\">\ubca1\ud130\uacf5\uac04\uacfc \uc120\ud615 \uad6c\uc870<\/a><\/li>\n<li><a href=\"..\/math-abstraction-07-topological-spaces\/\">\uac70\ub9ac\uacf5\uac04\uacfc \uc704\uc0c1\uacf5\uac04<\/a><\/li>\n<li><a href=\"..\/math-abstraction-08-measure-spaces\/\">\uce21\ub3c4\uacf5\uac04\uacfc \ud655\ub960\uacf5\uac04<\/a><\/li>\n<li><a href=\"..\/math-abstraction-09-cognitive-process\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \uc778\uc9c0\uc640 \ud559\uc2b5<\/a><\/li>\n<li><a href=\"..\/math-abstraction-10-optimal-generality\/\">\ud604\ub300 \uc218\ud559 \uc5f0\uad6c\uc5d0\uc11c\uc758 \ucd94\uc0c1\ud654<\/a><\/li>\n<li><a href=\"..\/math-abstraction-11-topography\/\">\uc218\ud559\uc801 \ucd94\uc0c1\ud654\uc758 \ubc29\ubc95\uacfc \ucca0\ud559<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><!-- class=\"math-abs-series\" --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>5\ubd80\uc758 \ub9d0\ubbf8\uc5d0\uc11c \uc608\uace0\ud588\ub4ef, \uc774\uc81c \uc2dc\uc120\uc744 \uad6c\uccb4\uc801\uc778 \uc218\ud559 \uac15\uc758\uc2e4\ub85c \uc62e\uaca8 \ubcf4\uc790. \uc5f0\ub9bd\uc77c\ucc28\ubc29\uc815\uc2dd, \uae30\ud558\ud559\uc758 \ubc29\ud5a5\ub7c9, \uc218\uc5f4, \ub2e4\ud56d\uc2dd, \ud568\uc218, \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \ud574\uc5d0\uc11c \uc624\ub298\ub0a0 \uc6b0\ub9ac\uac00 \u2018\uc120\ud615\uc131\u2019\uc774\ub77c\uace0 \ubd80\ub974\ub294 \uacc4\uc0b0\uc740 \uc11c\ub85c \ub2e4\ub978 \ub9e5\ub77d\uc5d0\uc11c \uc624\ub7ab\ub3d9\uc548 \uc0ac\uc6a9\ub418\uc5c8\ub2e4. \ud589\ub82c\uc2dd\uacfc \\(n\\)\ucc28\uc6d0 \uae30\ud558\ud559, \uadf8\ub77c\uc2a4\ub9cc\uc758 \ud655\uc7a5\ub860\ucc98\ub7fc \uc5ec\ub7ec \uc120\ud615 \ubb38\uc81c\ub97c \uc787\ub294 \uc2dc\ub3c4\ub3c4 \uc774\ubbf8 \uc9c4\ud589\ub418\uace0 \uc788\uc5c8\ub2e4. \ub2e4\ub9cc \uc774 \ud750\ub984\ub4e4\uc744 \uc720\ud55c\ucc28\uc6d0\uacfc \ubb34\ud55c\ucc28\uc6d0\uc5d0 \ub450\ub8e8 \uc801\uc6a9\ub418\ub294 \ud558\ub098\uc758 \uacf5\ub9ac\uc801 \uc5b8\uc5b4\ub85c \uc870\uc9c1\ud55c \ubca1\ud130\uacf5\uac04 \uc774\ub860\uc740 20\uc138\uae30 \uc804\ubc18\uc5d0 \uc774\ub974\ub7ec\uc11c\uc57c \ub110\ub9ac \uc790\ub9ac \uc7a1\uc558\ub2e4. \uc774 \uae00\uc5d0\uc11c\ub294 \uc774\ub7ec\ud55c \uc810\uc9c4\uc801\uc778&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9726,"menu_order":600,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9745","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9745","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9745"}],"version-history":[{"count":22,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9745\/revisions"}],"predecessor-version":[{"id":10053,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9745\/revisions\/10053"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9726"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9745"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}