{"id":9477,"date":"2025-10-20T18:35:11","date_gmt":"2025-10-20T09:35:11","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9477"},"modified":"2026-09-27T14:33:07","modified_gmt":"2026-09-27T05:33:07","slug":"ch02-metric-spaces","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\/","title":{"rendered":"\uac70\ub9ac\uacf5\uac04"},"content":{"rendered":"<div class=\"analysis2025\"><!-- ################## --><\/p>\n<p><!-- \n\n<h2>\uac70\ub9ac\uacf5\uac04<\/h2>\n\n --><\/p>\n<p><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\">1\uc7a5<\/a>\uc5d0\uc11c\ub294 \uc2e4\uc218\uc5f4\uc758 \uc218\ub834\uc744 \ub450 \uc2e4\uc218 \uc0ac\uc774\uc758 \uac70\ub9ac \\(|x-y|\\)\ub97c \uc774\uc6a9\ud558\uc5ec \uc815\uc758\ud558\uc600\ub2e4. \uac70\ub9ac\uacf5\uac04\uc740 \uc774\ub7ec\ud55c \u2018\uba40\uace0 \uac00\uae4c\uc6c0\u2019\uc758 \uac1c\ub150\uc744 \uc77c\ubc18\uc801\uc778 \uc9d1\ud569 \uc704\ub85c \ucd94\uc0c1\ud654\ud55c \uac83\uc774\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04\uc758 \uc815\uc758\uc640 \uc5ec\ub7ec \uc608\ub97c \uc0b4\ud3b4\ubcf4\uace0, \ubd80\ubd84\uacf5\uac04\uacfc \uac70\ub9ac\ub3d9\ud615\uc744 \uc815\uc758\ud55c\ub2e4.<\/p>\n<h3>\uac70\ub9ac\uacf5\uac04\uc758 \uc815\uc758<\/h3>\n<p>\uc9d1\ud569 \\(X\\)\uc640 \ud568\uc218 \\(d\\colon X\\times X\\to\\mathbb R\\)\uc774 \ub2e4\uc74c \uc138 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(d\\)\ub97c \\(X\\) \uc704\uc758 <span class=\"defined\">\uac70\ub9ac<\/span>(metric) \ub610\ub294 <span class=\"defined\">\uac70\ub9ac\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0, \\((X,d)\\)\ub97c <span class=\"defined\">\uac70\ub9ac\uacf5\uac04<\/span>(metric space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>(M1) \uc591\uc758 \uc815\ubd80\ud638\uc131: \\(d(x,y)=0\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(x=y\\)\uc774\ub2e4.<\/li>\n<li>(M2) \ub300\uce6d\uc131: \uc784\uc758\uc758 \\(x,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(d(x,y)=d(y,x)\\)\uc774\ub2e4.<\/li>\n<li>(M3) <span class=\"defined\">\uc0bc\uac01\ubd80\ub4f1\uc2dd<\/span>: \uc784\uc758\uc758 \\(x,y,z\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(d(x,z)\\leq d(x,y)+d(y,z)\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\uac70\ub9ac\ud568\uc218\ub97c \ud63c\ub3d9\ud560 \uc5fc\ub824\uac00 \uc5c6\uc744 \ub54c\ub294 \\((X,d)\\)\ub97c \uac04\ub2e8\ud788 \\(X\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc704 \uc815\uc758\uc5d0\uc11c\ub294 \uac70\ub9ac\uc758 \uacf5\uc5ed\uc744 \\([0,\\infty)\\)\ub85c \ubbf8\ub9ac \uc81c\ud55c\ud558\uc9c0 \uc54a\uc558\ub2e4. \uac70\ub9ac\uc758 \ube44\uc74c\uc218\uc131\uc740 \uc138 \uc870\uac74\uc5d0\uc11c \ub530\ub77c\uc624\uba70, \ub2e4\uc74c \ubb38\uc81c\uc5d0\uc11c \ud655\uc778\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p class=\"marginbottomhalf\"><span class=\"problem\">\ubb38\uc81c 2.1.<\/span><br \/>\n\uac70\ub9ac\uacf5\uac04 \\(X\\)\uc5d0 \uac70\ub9ac\ud568\uc218 \\(d\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<ol class=\"parenthesis\">\n<li>\uc784\uc758\uc758 \\(x,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(d(x,y)\\geq0\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(x,y,z\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(|d(x,z)-d(y,z)|\\leq d(x,y)\\)\uc774\ub2e4.<\/li>\n<li>\\(n\\geq2\\)\uc774\uace0 \\(x_1,x_2,\\ldots,x_n\\in X\\)\uc77c \ub54c,<br \/>\n\\[<br \/>\nd(x_1,x_n)\\leq d(x_1,x_2)+d(x_2,x_3)+\\cdots+d(x_{n-1},x_n)<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\uac70\ub9ac\uacf5\uac04 \\((X,d)\\)\uc5d0\uc11c \uc810 \\(x\\in X\\)\uc640 \uc591\uc218 \\(r>0\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\uc911\uc2ec\uc774 \\(x\\)\uc774\uace0 \ubc18\uc9c0\ub984\uc774 \\(r\\)\uc778 <span class=\"defined\">\uc5f4\ub9b0\uacf5<\/span>(open ball):<br \/>\n\\[<br \/>\nB(x,r)=\\{y\\in X\\mid d(x,y)&lt;r\\}<br \/>\n\\]<\/li>\n<li>\uc911\uc2ec\uc774 \\(x\\)\uc774\uace0 \ubc18\uc9c0\ub984\uc774 \\(r\\)\uc778 <span class=\"defined\">\ub2eb\ud78c\uacf5<\/span>(closed ball):<br \/>\n\\[<br \/>\n\\overline B(x,r)=\\{y\\in X\\mid d(x,y)\\leq r\\}<br \/>\n\\]<\/li>\n<li>\uc911\uc2ec\uc774 \\(x\\)\uc774\uace0 \ubc18\uc9c0\ub984\uc774 \\(r\\)\uc778 <span class=\"defined\">\uad6c\uba74<\/span>(sphere):<br \/>\n\\[<br \/>\nS(x,r)=\\{y\\in X\\mid d(x,y)=r\\}<br \/>\n\\]<\/li>\n<li>\uc911\uc2ec\uc774 \\(x\\)\uc774\uace0 \ubc18\uc9c0\ub984\uc774 \\(r\\)\uc778 <span class=\"defined\">\uad6c\uba4d\ub6ab\ub9b0 \uc5f4\ub9b0\uacf5<\/span>(punctured open ball):<br \/>\n\\[<br \/>\nB'(x,r)=\\{y\\in X\\mid 0&lt;d(x,y)&lt;r\\}<br \/>\n\\]<\/li>\n<\/ul>\n<p>\ucc45\uc5d0 \ub530\ub77c\uc11c\ub294 \uc704 \uc9d1\ud569\ub4e4\uc744 \uae30\ud638\ub85c \uac01\uac01 \\(B_r(x)\\), \\(\\overline B_r(x)\\), \\(S_r(x)\\), \\(B&#8217;_r(x)\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 2.2.<\/span><br \/>\n\\(B(x,r)\\)\uc774 \uc5f4\ub9b0\uacf5\uc774\uace0 \\(p\\in B(x,r)\\)\uc77c \ub54c, \\(B(p,\\delta)\\subseteq B(x,r)\\)\uc778 \\(\\delta>0\\)\uc774 \uc874\uc7ac\ud568\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774 \uc131\uc9c8\uc740 \ub4a4\uc5d0\uc11c \uc5f4\ub9b0\uc9d1\ud569\uc744 \ub2e4\ub8f0 \ub54c \uc0ac\uc6a9\ud55c\ub2e4.<!-- TODO: subsection:openandclosedsetsmetricspaces\uc758 \ucd5c\uc885 \uc808 \ubc88\ud638 \ubc0f \uc628\ub77c\uc778 \ub9c1\ud06c \ud655\uc778 --><\/p>\n<\/div>\n<h3>\uac70\ub9ac\uacf5\uac04\uc758 \uc608\uc2dc<\/h3>\n<h4>\uc720\ud074\ub9ac\ub4dc \uac70\ub9ac<\/h4>\n<p>\\(\\mathbb R^n\\)\uc5d0\uc11c <span class=\"defined\">\uc720\ud074\ub9ac\ub4dc \uac70\ub9ac<\/span>(Euclidean metric)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\nd_2(x,y)=\\sqrt{\\sum_{i=1}^{n}(x_i-y_i)^2}.<br \/>\n\\]<br \/>\n\ud2b9\ud788 \\(\\mathbb R\\)\uc5d0\uc11c\ub294 \\(d_2(x,y)=|x-y|\\)\uc774\uace0, \\(\\mathbb R^2\\)\uc5d0\uc11c\ub294 \\(d_2\\)\uac00 \uc77c\ubc18\uc801\uc778 \ud3c9\uba74\uc5d0\uc11c\uc758 \uac70\ub9ac\uac00 \ub41c\ub2e4.<\/p>\n<p>\ub354 \uc77c\ubc18\uc801\uc73c\ub85c, \\(1\\leq p&lt;\\infty\\)\uc77c \ub54c \\(\\mathbb R^n\\)\uc5d0\uc11c <span class=\"defined\">\\(p\\)-\uac70\ub9ac<\/span>\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\nd_p(x,y)=\\left(\\sum_{i=1}^{n}|x_i-y_i|^p\\right)^{1\/p}.<br \/>\n\\]<br \/>\n\uc77c\ubc18\uc801\uc778 \\(p\\)\uc5d0 \ub300\ud558\uc5ec \uc0bc\uac01\ubd80\ub4f1\uc2dd\uc744 \ubcf4\uc774\ub294 \ud575\uc2ec\uc740 \ub2e4\uc74c <span class=\"defined\">\ubbfc\ucf54\ud504\uc2a4\ud0a4 \ubd80\ub4f1\uc2dd<\/span>(Minkowski&#8217;s inequality)\uc774\ub2e4.<br \/>\n\\[<br \/>\n\\left(\\sum_{i=1}^n|u_i+v_i|^p\\right)^{1\/p}<br \/>\n\\leq<br \/>\n\\left(\\sum_{i=1}^n|u_i|^p\\right)^{1\/p}<br \/>\n+<br \/>\n\\left(\\sum_{i=1}^n|v_i|^p\\right)^{1\/p}.<br \/>\n\\]<br \/>\n\uc774 \ubd80\ub4f1\uc2dd\uc758 \uc99d\uba85\uc740 \uc774 \uc7a5\uc758 \ubc94\uc704\ub97c \ubc97\uc5b4\ub098\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n<p>\ub610\ud55c \\(p=\\infty\\)\uc5d0 \ub300\uc751\ud558\ub294 \uac70\ub9ac\ub85c<br \/>\n\\[<br \/>\nd_\\infty(x,y)=\\max_{1\\leq i\\leq n}|x_i-y_i|<br \/>\n\\]<br \/>\n\ub97c \uc815\uc758\ud55c\ub2e4. \uc774 \uac70\ub9ac\ub97c <span class=\"defined\">\ucd5c\ub300 \uac70\ub9ac<\/span> \ub610\ub294 <span class=\"defined\">\uade0\ub4f1 \uac70\ub9ac<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 2.3.<\/span><br \/>\n\uc720\ud074\ub9ac\ub4dc \uac70\ub9ac, \\(p\\)-\uac70\ub9ac, \uade0\ub4f1\uac70\ub9ac\uac00 \ubaa8\ub450 \uac70\ub9ac\ud568\uc218\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc774\uc2dc\uc624. \uc77c\ubc18\uc801\uc778 \\(1&lt;p&lt;\\infty\\)\uc758 \uacbd\uc6b0\uc5d0\ub294 \uc704\uc758 \ubbfc\ucf54\ud504\uc2a4\ud0a4 \ubd80\ub4f1\uc2dd\uc744 \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/p>\n<\/div>\n<p>\\(\\mathbb R^2\\)\uc5d0\uc11c \uc810 \\(O=(0,0)\\)\uc744 \uc911\uc2ec\uc73c\ub85c \ud558\uace0 \ubc18\uc9c0\ub984\uc774 1\uc778 \uc5f4\ub9b0\uacf5\uc758 \ubaa8\uc591\uc740 \uac70\ub9ac\ud568\uc218\uc5d0 \ub530\ub77c \ub2e4\ub974\ub2e4.<\/p>\n<ul>\n<li>\uac70\ub9ac\ud568\uc218\uac00 \\(d_1\\)\uc77c \ub54c \\(B(O,1)\\)\uc740 \ub9c8\ub984\ubaa8 \ubaa8\uc591\uc774\ub2e4.<\/li>\n<li>\uac70\ub9ac\ud568\uc218\uac00 \\(d_2\\)\uc77c \ub54c \\(B(O,1)\\)\uc740 \uc6d0 \ubaa8\uc591\uc774\ub2e4.<\/li>\n<li>\uac70\ub9ac\ud568\uc218\uac00 \\(d_\\infty\\)\uc77c \ub54c \\(B(O,1)\\)\uc740 \uc815\uc0ac\uac01\ud615 \ubaa8\uc591\uc774\ub2e4.<\/li>\n<\/ul>\n<h4>\ubcf5\uc18c\ud3c9\uba74<\/h4>\n<p><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\">\ubb38\uc81c 1.13<\/a>\uc5d0\uc11c\ucc98\ub7fc \ubcf5\uc18c\uc218 \\(z=a+bi\\)\ub97c \uc810 \\((a,b)\\in\\mathbb R^2\\)\uc640 \ub300\uc751\uc2dc\ud0a4\uc790. \\(|z|=\\sqrt{a^2+b^2}\\)\ub85c \uc815\uc758\ud558\uace0<br \/>\n\\[<br \/>\nd(z,w)=|z-w|<br \/>\n\\]<br \/>\n\ub85c \ub450\uba74 \\(\\mathbb C\\)\ub294 \uac70\ub9ac\uacf5\uac04\uc774 \ub41c\ub2e4. \uc2e4\uc81c\ub85c \\(z=a+bi\\), \\(w=c+di\\)\uc774\uba74<br \/>\n\\[<br \/>\n|z-w|=\\sqrt{(a-c)^2+(b-d)^2},<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \uc774 \uac70\ub9ac\ub294 \\(\\mathbb R^2\\)\uc758 \uc720\ud074\ub9ac\ub4dc \uac70\ub9ac\uc640 \uc815\ud655\ud788 \ub300\uc751\ud55c\ub2e4.<\/p>\n<h4>\uc774\uc0b0\uac70\ub9ac\uacf5\uac04<\/h4>\n<p>\uc784\uc758\uc758 \uc9d1\ud569 \\(X\\)\uc5d0 \ub300\ud558\uc5ec <span class=\"defined\">\uc774\uc0b0\uac70\ub9ac<\/span>(discrete metric)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[<br \/>\nd(x,y)=<br \/>\n\\begin{cases}<br \/>\n0 &#038; \\text{if }\\;x=y,\\\\[5pt]<br \/>\n1 &#038; \\text{if }\\;x\\neq y.<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\uc0b0\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c\ub294 \uac01 \uc810\uc744 \ucda9\ubd84\ud788 \uc791\uc740 \uc5f4\ub9b0\uacf5 \ud558\ub098\ub85c \ub530\ub85c \ub5bc\uc5b4\ub0bc \uc218 \uc788\ub2e4. \uc2e4\uc81c\ub85c \\(0&lt;r\\leq1\\)\uc77c \ub54c \\(B(x,r)=\\{x\\}\\)\uc774\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 2.4.<\/span><br \/>\n\uc774\uc0b0\uac70\ub9ac\uac00 \uac70\ub9ac\ud568\uc218\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<h4>\ud568\uc218\uacf5\uac04<\/h4>\n<p>\\(E\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(f\\colon E\\to\\mathbb R\\)\uc5d0 \ub300\ud558\uc5ec \uc5b4\ub5a4 \\(M>0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \ubaa8\ub4e0 \\(x\\in E\\)\uc5d0\uc11c \\(|f(x)|\\leq M\\)\uc774\uba74 \\(f\\)\uac00 \\(E\\)\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud55c\ub2e4. \\(E\\)\uc5d0\uc11c \uc720\uacc4\uc778 \uc2e4\ud568\uc218\ub4e4\uc758 \uc9d1\ud569\uc744 \\(B(E)\\)\ub77c\uace0 \ud558\uc790. \\(f,g\\in B(E)\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nd_\\infty(f,g)=\\sup_{x\\in E}|f(x)-g(x)|<br \/>\n\\]<br \/>\n\ub85c \ub450\uba74 \\(d_\\infty\\)\ub294 \\(B(E)\\) \uc704\uc758 \uac70\ub9ac\ud568\uc218\uac00 \ub41c\ub2e4. \\(f-g\\)\uac00 \uc720\uacc4\uc774\ubbc0\ub85c \uc704 \uc0c1\ud55c\uc740 \uc720\ud55c\ud55c \uc2e4\uc218\ub85c \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\ub4a4\uc5d0\uc11c \uc5f0\uc18d\ud568\uc218\ub97c \uc815\uc758\ud558\uba74 \ub2eb\ud78c\uad6c\uac04 \\([a,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \uc2e4\ud568\uc218\ub4e4\uc758 \uc9d1\ud569 \\(C[a,b]\\)\ub3c4 \uc774 \uac70\ub9ac\uc758 \uc911\uc694\ud55c \uc608\uac00 \ub41c\ub2e4. \uc801\ubd84\uc744 \uc815\uc758\ud55c \ub4a4\uc5d0\ub294 \\(L^p\\) \ud615\ud0dc\uc758 \uac70\ub9ac\ub3c4 \ub2e4\ub8f0 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 2.5.<\/span><br \/>\n\\(E\\neq\\varnothing\\)\uc774\uace0 \\(B(E)\\)\uac00 \\(E\\)\uc5d0\uc11c \uc720\uacc4\uc778 \uc2e4\ud568\uc218\ub4e4\uc758 \uc9d1\ud569\uc77c \ub54c, \uc704\uc5d0\uc11c \uc815\uc758\ud55c \uade0\ub4f1\uac70\ub9ac \\(d_\\infty\\)\uac00 \uac70\ub9ac\ud568\uc218\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<h4>\ub178\ub984\uacf5\uac04<\/h4>\n<p>\uc120\ud615\ub300\uc218\ud559\uc5d0\uc11c \ub2e4\ub8e8\ub294 \uc2e4\ubca1\ud130\uacf5\uac04 \\(V\\)\ub97c \uc0dd\uac01\ud558\uc790. \ud568\uc218 \\(\\lVert\\cdot\\rVert\\colon V\\to[0,\\infty)\\)\uac00 \ub2e4\uc74c \uc138 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \uc774\ub97c <span class=\"defined\">\ub178\ub984<\/span>(norm)\uc774\ub77c\uace0 \ud558\uace0, \\((V,\\lVert\\cdot\\rVert)\\)\ub97c <span class=\"defined\">\ub178\ub984\uacf5\uac04<\/span>(normed space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>(N1) \\(\\lVert u\\rVert=0\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(u=0\\)\uc774\ub2e4.<\/li>\n<li>(N2) \uc784\uc758\uc758 \\(\\alpha\\in\\mathbb R\\), \\(u\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lVert\\alpha u\\rVert=|\\alpha|\\lVert u\\rVert\\)\uc774\ub2e4.<\/li>\n<li>(N3) \uc784\uc758\uc758 \\(u,v\\in V\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lVert u+v\\rVert\\leq\\lVert u\\rVert+\\lVert v\\rVert\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<p>\ub178\ub984\uacf5\uac04\uc5d0\uc11c<br \/>\n\\[<br \/>\nd(u,v)=\\lVert u-v\\rVert<br \/>\n\\]<br \/>\n\ub85c \uc815\uc758\ud558\uba74 \\(d\\)\ub294 \uac70\ub9ac\ud568\uc218\uac00 \ub41c\ub2e4. \ub530\ub77c\uc11c \ubaa8\ub4e0 \ub178\ub984\uacf5\uac04\uc740 \uc790\uc5f0\uc2a4\ub7fd\uac8c \uac70\ub9ac\uacf5\uac04\uc774 \ub41c\ub2e4. \uc608\ub97c \ub4e4\uc5b4 \\(\\mathbb R^n\\)\uc5d0\uc11c<br \/>\n\\[<br \/>\n\\|x\\|_p=\\left(\\sum_{i=1}^n|x_i|^p\\right)^{1\/p},<br \/>\n\\qquad<br \/>\n\\|x\\|_\\infty=\\max_{1\\leq i\\leq n}|x_i|<br \/>\n\\]<br \/>\n\ub85c \ub450\uba74<br \/>\n\\[<br \/>\nd_p(x,y)=\\|x-y\\|_p,\\qquad<br \/>\nd_\\infty(x,y)=\\|x-y\\|_\\infty<br \/>\n\\]<br \/>\n\uc774\ub2e4.<\/p>\n<h3>\ubd80\ubd84\uacf5\uac04\uacfc \uac70\ub9ac\ub3d9\ud615<\/h3>\n<p>\uac70\ub9ac\uacf5\uac04 \\((X,d)\\)\uc758 \ubd80\ubd84\uc9d1\ud569 \\(Y\\subseteq X\\)\uc5d0 \ub300\ud558\uc5ec, \\(d\\)\uc758 \uc815\uc758\uc5ed\uc744 \\(Y\\times Y\\)\ub85c \uc81c\ud55c\ud55c \ud568\uc218 \\(d_Y\\)\ub294 \\(Y\\) \uc704\uc758 \uac70\ub9ac\uac00 \ub41c\ub2e4. \uc774\ub54c \uac70\ub9ac\uacf5\uac04 \\((Y,d_Y)\\)\ub97c \\((X,d)\\)\uc758 <span class=\"defined\">\ubd80\ubd84\uac70\ub9ac\uacf5\uac04<\/span>(metric subspace) \ub610\ub294 \uac04\ub2e8\ud788 <span class=\"defined\">\ubd80\ubd84\uacf5\uac04<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\uc608\ub97c \ub4e4\uc5b4, \uad6c\uac04 \\([0,1]\\)\uc740 \\(\\mathbb R\\)\uc758 \ubd80\ubd84\uac70\ub9ac\uacf5\uac04\uc774\uace0, \ub2e8\uc704\uc6d0<br \/>\n\\[<br \/>\nS^1=\\{z\\in\\mathbb C\\mid |z|=1\\}<br \/>\n\\]<br \/>\n\uc740 \\(\\mathbb C\\)\uc758 \ubd80\ubd84\uac70\ub9ac\uacf5\uac04\uc774\ub2e4.<\/p>\n<p>\ub450 \uac70\ub9ac\uacf5\uac04 \\((X,d_X)\\)\uc640 \\((Y,d_Y)\\) \uc0ac\uc774\uc758 \ud568\uc218 \\(f\\colon X\\to Y\\)\uac00<br \/>\n\\[<br \/>\nd_Y(f(x_1),f(x_2))=d_X(x_1,x_2)<br \/>\n\\]<br \/>\n\ub97c \ubaa8\ub4e0 \\(x_1,x_2\\in X\\)\uc5d0 \ub300\ud558\uc5ec \ub9cc\uc871\uc2dc\ud0ac \ub54c \\(f\\)\ub97c <span class=\"defined\">\uac70\ub9ac\ubcf4\uc874\ud568\uc218<\/span>(isometry)\ub77c\uace0 \ubd80\ub978\ub2e4. \uac70\ub9ac\ubcf4\uc874\ud568\uc218\ub294 \uc77c\ub300\uc77c\ud568\uc218\uc774\ub2e4. \uc2e4\uc81c\ub85c \\(f(x_1)=f(x_2)\\)\uc774\uba74<br \/>\n\\[<br \/>\nd_X(x_1,x_2)=d_Y(f(x_1),f(x_2))=0<br \/>\n\\]<br \/>\n\uc774\ubbc0\ub85c \\(x_1=x_2\\)\uc774\ub2e4. \uac70\ub9ac\ubcf4\uc874\ud568\uc218 \\(f\\colon X\\to Y\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc774\uba74 \ub450 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc640 \\(Y\\)\uac00 <span class=\"defined\">\uac70\ub9ac\ub3d9\ud615<\/span>\uc774\ub2e4(isometric)\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 2.6.<\/span><br \/>\n\ud568\uc218 \\(f\\colon\\mathbb R\\to\\mathbb R^2\\)\ub97c<br \/>\n\\[<br \/>\nf(t)=(t,t^2)<br \/>\n\\]<br \/>\n\uc73c\ub85c \uc815\uc758\ud560 \ub54c, \\(f\\)\uac00 \uac70\ub9ac\ubcf4\uc874\ud568\uc218\uac00 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 2.7.<\/span><br \/>\n\\(\\mathbb R^2\\)\uc5d0 \uc720\ud074\ub9ac\ub4dc \uac70\ub9ac\uac00 \uc8fc\uc5b4\uc838 \uc788\ub2e4\uace0 \ud558\uc790. \uc120\ud615\ubcc0\ud658 \\(L\\colon\\mathbb R^2\\to\\mathbb R^2\\)\uac00 \uac70\ub9ac\ub97c \ubcf4\uc874\ud558\uba74 \\(L\\)\uc758 \ud589\ub82c\uc758 \ub450 \uc5f4\ubca1\ud130\uac00 \uc815\uaddc\uc9c1\uad50\uae30\uc800\ub97c \uc774\ub8f8\uc744 \ubcf4\uc774\uc2dc\uc624. \uc774\ub97c \uc774\uc6a9\ud558\uc5ec \\(L\\)\uc774 \ud68c\uc804\ubcc0\ud658 \ub610\ub294 \uc6d0\uc810\uc744 \uc9c0\ub098\ub294 \uc9c1\uc120\uc5d0 \ub300\ud55c \ub300\uce6d\ubcc0\ud658\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ub450 \uac70\ub9ac\uacf5\uac04 \\((X,d_X)\\)\uc640 \\((Y,d_Y)\\)\uc758 \uacf1\uc9d1\ud569 \\(X\\times Y\\)\uc5d0\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc5ec\ub7ec \uac70\ub9ac\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li><span class=\"defined\">\uc720\ud074\ub9ac\ub4dc \uacf1\uac70\ub9ac<\/span>:<br \/>\n\\[<br \/>\nd_2((x_1,y_1),(x_2,y_2))<br \/>\n=\\sqrt{d_X(x_1,x_2)^2+d_Y(y_1,y_2)^2}<br \/>\n\\]<\/li>\n<li><span class=\"defined\">\ud0dd\uc2dc \uacf1\uac70\ub9ac<\/span>:<br \/>\n\\[<br \/>\nd_1((x_1,y_1),(x_2,y_2))<br \/>\n=d_X(x_1,x_2)+d_Y(y_1,y_2)<br \/>\n\\]<\/li>\n<li><span class=\"defined\">\ucd5c\ub300 \uacf1\uac70\ub9ac<\/span>:<br \/>\n\\[<br \/>\nd_\\infty((x_1,y_1),(x_2,y_2))<br \/>\n=\\max\\{d_X(x_1,x_2),d_Y(y_1,y_2)\\}<br \/>\n\\]<\/li>\n<\/ul>\n<p>\uc138 \uac70\ub9ac \uc911 \ud558\ub098\ub97c \uc900 \uac70\ub9ac\uacf5\uac04 \\((X\\times Y,d)\\)\ub97c <span class=\"defined\">\uacf1\uac70\ub9ac\uacf5\uac04<\/span>(product metric space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\ub4e4\uc774 \ubaa8\ub450 \uac70\ub9ac\ud568\uc218\uc784\uc740 \ub2e4\uc74c \ubb38\uc81c\uc5d0\uc11c \ud655\uc778\ud55c\ub2e4. \ub610\ud55c \ubaa8\ub4e0 \\(p,q\\in X\\times Y\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[<br \/>\nd_\\infty(p,q)\\leq d_2(p,q)\\leq d_1(p,q)\\leq2d_\\infty(p,q)<br \/>\n\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\ubbc0\ub85c \uc138 \uac70\ub9ac\ub294 \uc11c\ub85c \uc815\ub7c9\uc801\uc73c\ub85c \ube44\uad50\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 2.8.<\/span><br \/>\n\uc720\ud074\ub9ac\ub4dc \uacf1\uac70\ub9ac, \ud0dd\uc2dc \uacf1\uac70\ub9ac, \ucd5c\ub300 \uacf1\uac70\ub9ac\uac00 \ubaa8\ub450 \uac70\ub9ac\ud568\uc218\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<div class=\"contentbottombox\">\n<p class=\"contentbottomboxtitle\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/\">\ud574\uc11d\ud559 \uac15\uc758\ub178\ud2b8<\/a><\/p>\n<ol class=\"contentboxorderedlist\">\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch01-real-number-system\">\uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/a><\/li>\n<li class=\"contentboxthis\"><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch02-metric-spaces\">\uac70\ub9ac\uacf5\uac04<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch03-limit-of-sequences\">\uc218\uc5f4\uc758 \uadf9\ud55c\uacfc \uc704\uc0c1\uc801 \uc131\uc9c8<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch04-limit-of-functions-and-continuity\">\ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch05-differentiation-of-functions-of-one-variable\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch06-the-riemann-integral\">\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch07-infinite-series\">\ubb34\ud55c\uae09\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch08-real-analytic-functions\">\uc2e4\ud574\uc11d\uc801 \ud568\uc218<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch09-differentiation-of-functions-of-several-variables\">\ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\">\uc911\uc801\ubd84<\/a><\/li>\n<li><a href=\"\/blog\/invitation-to-mathematical-analysis\/ch11-vector-field-and-fundamental-theorems\">\ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><!-- ################## --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>1\uc7a5\uc5d0\uc11c\ub294 \uc2e4\uc218\uc5f4\uc758 \uc218\ub834\uc744 \ub450 \uc2e4\uc218 \uc0ac\uc774\uc758 \uac70\ub9ac \\(|x-y|\\)\ub97c \uc774\uc6a9\ud558\uc5ec \uc815\uc758\ud558\uc600\ub2e4. \uac70\ub9ac\uacf5\uac04\uc740 \uc774\ub7ec\ud55c \u2018\uba40\uace0 \uac00\uae4c\uc6c0\u2019\uc758 \uac1c\ub150\uc744 \uc77c\ubc18\uc801\uc778 \uc9d1\ud569 \uc704\ub85c \ucd94\uc0c1\ud654\ud55c \uac83\uc774\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04\uc758 \uc815\uc758\uc640 \uc5ec\ub7ec \uc608\ub97c \uc0b4\ud3b4\ubcf4\uace0, \ubd80\ubd84\uacf5\uac04\uacfc \uac70\ub9ac\ub3d9\ud615\uc744 \uc815\uc758\ud55c\ub2e4. \uac70\ub9ac\uacf5\uac04\uc758 \uc815\uc758 \uc9d1\ud569 \\(X\\)\uc640 \ud568\uc218 \\(d\\colon X\\times X\\to\\mathbb R\\)\uc774 \ub2e4\uc74c \uc138 \uc870\uac74\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(d\\)\ub97c \\(X\\) \uc704\uc758 \uac70\ub9ac(metric) \ub610\ub294 \uac70\ub9ac\ud568\uc218\ub77c\uace0 \ubd80\ub974\uace0, \\((X,d)\\)\ub97c \uac70\ub9ac\uacf5\uac04(metric space)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. (M1) \uc591\uc758 \uc815\ubd80\ud638\uc131: \\(d(x,y)=0\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(x=y\\)\uc774\ub2e4. (M2)&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":102,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9477","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9477","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=9477"}],"version-history":[{"count":9,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9477\/revisions"}],"predecessor-version":[{"id":10106,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9477\/revisions\/10106"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9470"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9477"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}