{"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":"2025-10-21T16:07:36","modified_gmt":"2025-10-21T07:07:36","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>\uac70\ub9ac\uacf5\uac04\uc740 &#8216;\uba40\uace0 \uac00\uae4c\uc6c0\uc744 \uc7b4 \uc218 \uc788\ub294 \uacf5\uac04&#8217;\uc758 \uac1c\ub150\uc744 \ucd94\uc0c1\ud654\ud55c \uac83\uc73c\ub85c, \ud574\uc11d\ud559\uacfc \uc704\uc0c1\uc218\ud559\uc758 \uae30\ucd08\uac00 \ub41c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04\uc758 \uc815\uc758\uc640 \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n<h3>\uac70\ub9ac\uacf5\uac04\uc758 \uc815\uc758<\/h3>\n<p>\uc9d1\ud569 \\(X\\)\uc640 \ud568\uc218 \\(d: 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>\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>\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><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.<\/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) \\geq 0\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(x,\\,y,\\,z\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(|d(x,\\, z) &#8211; d(y,\\, z)| \\leq d(x,\\, y)\\)\uc774\ub2e4.<\/li>\n<li>\uc784\uc758\uc758 \\(x_1\\), \\(x_2\\), \\(\\cdots\\), \\(x_n\\in X\\)\uc5d0 \ub300\ud558\uc5ec, \\(d(x_1,\\, x_n) \\leq d(x_1,\\, x_2) + d(x_2,\\, x_3) + \\cdots + d(x_{n-1},\\, x_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): \\(B(x,\\, r) = \\{y \\in X \\mid d(x,\\, y) < r\\}\\)<\/li>\n<li>\uc911\uc2ec\uc774 \\(x\\)\uc774\uace0 \ubc18\uc9c0\ub984\uc774 \\(r\\)\uc778 <span class=\"defined\">\ub2eb\ud78c\uacf5<\/span>(closed ball): \\(\\overline{B}(x,\\, r) = \\{y \\in X \\mid d(x,\\, y) \\leq r\\}\\)<\/li>\n<li>\uc911\uc2ec\uc774 \\(x\\)\uc774\uace0 \ubc18\uc9c0\ub984\uc774 \\(r\\)\uc778 <span class=\"defined\">\uad6c\uba74<\/span>(sphere): \\(S(x,\\, r) = \\{y \\in X \\mid d(x,\\, y) = r\\}\\)<\/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>: \\(B&#8217; (x,\\, r) = \\{y \\in X \\mid 0 < d(x,\\, y) < r\\}\\)<\/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_r &#8216; (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 \uba85\uc81c\ub294 \uc5f4\ub9b0\uacf5\uc774 \uc2e4\uc81c\ub85c \uc5f4\ub9b0\uc9d1\ud569\uc784\uc744 \uc124\uba85\ud55c\ub2e4.)<\/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\\[d_2(x,\\, y) = \\sqrt{\\sum_{i=1}^{n} (x_i &#8211; y_i)^2}.\\]<\/p>\n<p>\ud2b9\ud788 \\(\\mathbb{R}\\)\uc5d0\uc11c\ub294 \\(d_2 (x,\\, y) = |x &#8211; 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, \\(p\\ge 1\\)\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\\[d_p(x,\\, y) = \\left(\\sum_{i=1}^{n} |x_i &#8211; y_i|^p\\right)^{1\/p}.\\]<\/p>\n<p>\\(p = \\infty\\)\uc77c \ub54c \uc704 \uac70\ub9ac\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \ubc14\ub010\ub2e4.<br \/>\n\\[d_\\infty(x,\\, y) = \\max_{1 \\leq i \\leq n} \\left| x_i &#8211; y_i \\right|.\\]<br \/>\n\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.<\/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>\ubcf5\uc18c\uc218 \uc9d1\ud569 \\(\\mathbb{C}\\)\uc5d0\uc11c \uac70\ub9ac\ub97c \\(d(z,\\, w) = |z &#8211; w|\\)\ub77c\uace0 \uc815\uc758\ud558\uba74 \\(\\mathbb{C}\\)\ub294 \uac70\ub9ac\uacf5\uac04\uc774 \ub41c\ub2e4. \uc5ec\uae30\uc11c \\(|z|\\)\ub294 \ubcf5\uc18c\uc218\uc758 \uc808\ub313\uac12\uc774\ub2e4. \uc774 \uac70\ub9ac\uacf5\uac04\uc740 \\(\\mathbb{R}^2\\)\uc758 \uc720\ud074\ub9ac\ub4dc \uac70\ub9ac\uacf5\uac04\uacfc \uac70\ub9ac\ub3d9\ud615\uc774\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\\[d(x,\\,y) =<br \/>\n\\begin{cases}<br \/>\n0 &#038; \\text{if }\\, x = y, \\\\[6pt]<br \/>\n1 &#038; \\text{if }\\, x \\neq y.<br \/>\n\\end{cases}\\]<\/p>\n<p>\uc774\uc0b0\uac70\ub9ac\uacf5\uac04\uc5d0\uc11c\ub294 \ubaa8\ub4e0 \uc810\uc774 &#8216;\uace0\ub9bd&#8217;\ub418\uc5b4 \uc788\ub2e4. \uc989 \\(r\\)\uc774 \\(1\\)\ubcf4\ub2e4 \uc791\uc740 \uc591\uc218\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>\uc5f0\uc18d\ud568\uc218\uacf5\uac04<\/h4>\n<p>\uad6c\uac04 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \uc2e4\ud568\uc218\ub4e4\uc758 \uc9d1\ud569\uc744 \\(C[a,\\, b]\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774 \uc9d1\ud569\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc740 \uac70\ub9ac\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li>\uade0\ub4f1\uac70\ub9ac: \\(d_\\infty(f,\\, g) = \\displaystyle\\sup_{x \\in [a,\\, b]} |f(x) &#8211; g(x)|\\)<\/li>\n<li>\\(L^p\\) \uac70\ub9ac: \\(d_p(f,\\, g) = \\displaystyle\\left(\\int_a^b |f(x) &#8211; g(x)|^p \\, dx\\right)^{1\/p}\\) (\ub2e8, \\(p\\ge 1\\))<\/li>\n<\/ul>\n<div class=\"problem\">\n<p><span class=\"problem\">\ubb38\uc81c 2.5.<\/span><br \/>\n\uade0\ub4f1\uac70\ub9ac\uac00 \uac70\ub9ac\ud568\uc218\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc774\uc2dc\uc624. \ud2b9\ud788 \uc9d1\ud569 \\(E\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f:E\\rightarrow \\mathbb{R}\\)\uc758 \ubaa8\uc784\uc744 \\(X\\)\ub77c\uace0 \ud588\uc744 \ub54c, \\(X\\)\uc5d0\uc11c\ub3c4 \uade0\ub4f1\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>\ub178\ub984\uacf5\uac04<\/h4>\n<p>\ubca1\ud130\uacf5\uac04 \\(V\\)\uc5d0 \ub178\ub984 \\(\\lVert \\cdot \\rVert\\)\uc774 \uc8fc\uc5b4\uc838 \uc788\uc744 \ub54c, \\(u,\\,v\\in V\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[d(u,\\,v) = \\lVert u-v \\rVert\\]<br \/>\n\ub77c\uace0 \ud558\uba74, \\(d\\)\ub294 \\(V\\) \uc704\uc758 \uac70\ub9ac\ud568\uc218\uac00 \ub41c\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \ub178\ub984\uacf5\uac04\uc740 \ubaa8\ub450 \uac70\ub9ac\uacf5\uac04\uc774\ub2e4.<\/p>\n<p>\ub9c8\ucc2c\uac00\uc9c0\ub85c \ubca1\ud130\uacf5\uac04 \\(V\\)\uc5d0 \ub0b4\uc801 \\(\\langle \\cdot ,\\, \\cdot \\rangle\\)\uc774 \uc8fc\uc5b4\uc838 \uc788\uc744 \ub54c, \\(u\\in V\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\lVert u \\rVert = \\sqrt{\\langle u ,\\,u \\rangle}\\]<br \/>\n\ub77c\uace0 \ud558\uba74 \\(\\lVert \\cdot \\rVert\\)\ub294 \\(V\\)\uc758 \ub178\ub984\uc774 \ub41c\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \ub0b4\uc801\uacf5\uac04\uc740 \ub178\ub984\uacf5\uac04\uc774\uba70, \ub0b4\uc801\uacf5\uac04\uc740 \uac70\ub9ac\uacf5\uac04\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 \\(S^1 = \\{z \\in \\mathbb{C} \\mid |z| = 1\\}\\)\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: X \\to Y\\)\uac00 \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c \\(f\\)\ub97c <span class=\"defined\">\uac70\ub9ac\ubcf4\uc874\ud568\uc218<\/span>(isometry)\ub77c\uace0 \ubd80\ub978\ub2e4.<br \/>\n\\[\\text{\uc784\uc758\uc758 }x_1 ,\\, x_2 \\in X \\text{\uc5d0 \ub300\ud558\uc5ec, }\\, d_Y(f(x_1),\\, f(x_2)) = d_X(x_1,\\, x_2).\\]<br \/>\n\uac70\ub9ac\ubcf4\uc874\ud568\uc218\ub294 \uac70\ub9ac\ub97c \ubcf4\uc874\ud558\ubbc0\ub85c \uc77c\ub300\uc77c\ud568\uc218\uac00 \ub41c\ub2e4. \ub9cc\uc57d \uac70\ub9ac\ubcf4\uc874\ud568\uc218 \\(f:X\\rightarrow Y\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc774\uba74 &#8220;\ub450 \uac70\ub9ac\uacf5\uac04 \\(X\\)\uc640 \\(Y\\)\uac00 <span class=\"defined\">\uac70\ub9ac\ub3d9\ud615<\/span>\uc774\ub2e4(isometric)&#8221;\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: \\mathbb{R} \\to \\mathbb{R}^2\\)\ub97c \\(f(t) = \\left(t,\\, t^2\\right)\\)\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\\)\ub97c \ubcf4\ud1b5\uac70\ub9ac\uac00 \uc8fc\uc5b4\uc9c4 2\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(L : \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2\\)\uac00 \uac70\ub9ac\ub97c \ubcf4\uc874\ud558\ub294 \uc120\ud615\ubcc0\ud658\uc774\uba74, \\(L\\)\uc740 \ud68c\uc804\ubcc0\ud658\uc774\uac70\ub098 \ub300\uce6d\ubcc0\ud658\uc774\uac70\ub098 \ub610\ub294 \ud68c\uc804\ubcc0\ud658\uacfc \ub300\uce6d\ubcc0\ud658\uc758 \ud569\uc131\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n<p>\ub450 \uac70\ub9ac\uacf5\uac04 \\((X,\\, d_X)\\)\uc640 \\((Y,\\, d_Y)\\)\uc758 <span class=\"defined\">\uacf1\uac70\ub9ac\uacf5\uac04<\/span>(product metric space) \\((X \\times Y,\\, d)\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ul>\n<li>\uc720\ud074\ub9ac\ub4dc \uacf1\uac70\ub9ac: \\(d((x_1,\\, y_1),\\, (x_2,\\, y_2)) = \\sqrt{d_X(x_1,\\, x_2)^2 + d_Y(y_1,\\, y_2)^2}\\)<\/li>\n<li>\ud0dd\uc2dc \uacf1\uac70\ub9ac: \\(d((x_1,\\, y_1),\\, (x_2,\\, y_2)) = d_X(x_1,\\, x_2) + d_Y(y_1,\\, y_2)\\)<\/li>\n<li>\ucd5c\ub300 \uacf1\uac70\ub9ac: \\(d((x_1,\\, y_1),\\, (x_2,\\, y_2)) = \\max\\{d_X(x_1,\\, x_2),\\, d_Y(y_1,\\, y_2)\\}\\)<\/li>\n<\/ul>\n<p>\uc774\ub4e4\uc740 \ubaa8\ub450 \uac70\ub9ac\ud568\uc218\uc758 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba70, \uc774 \uac70\ub9ac\ub4e4\uc5d0 \uc758\ud574 \uc815\uc758\ub41c \uacf1\uacf5\uac04\ub4e4\uc740 \uc11c\ub85c \uc704\uc0c1\ub3d9\ud615\uc774\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 \ud575\uc2ec\uc815\ub9ac \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<\/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>\uac70\ub9ac\uacf5\uac04\uc740 &#8216;\uba40\uace0 \uac00\uae4c\uc6c0\uc744 \uc7b4 \uc218 \uc788\ub294 \uacf5\uac04&#8217;\uc758 \uac1c\ub150\uc744 \ucd94\uc0c1\ud654\ud55c \uac83\uc73c\ub85c, \ud574\uc11d\ud559\uacfc \uc704\uc0c1\uc218\ud559\uc758 \uae30\ucd08\uac00 \ub41c\ub2e4. \uc774 \uc7a5\uc5d0\uc11c\ub294 \uac70\ub9ac\uacf5\uac04\uc758 \uc815\uc758\uc640 \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uac70\ub9ac\uacf5\uac04\uc758 \uc815\uc758 \uc9d1\ud569 \\(X\\)\uc640 \ud568\uc218 \\(d: 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. \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. \ub300\uce6d\uc131: \uc784\uc758\uc758 \\(x,\\,y\\in&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":7,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9477\/revisions"}],"predecessor-version":[{"id":9607,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9477\/revisions\/9607"}],"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}]}}