{"id":2584,"date":"2019-03-05T14:57:27","date_gmt":"2019-03-05T05:57:27","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=2584"},"modified":"2025-10-20T15:35:54","modified_gmt":"2025-10-20T06:35:54","slug":"integration-of-abstract-functions","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/integration-of-abstract-functions\/","title":{"rendered":"\ucd94\uc0c1\uacf5\uac04\uc5d0\uc11c\uc758 \uc801\ubd84"},"content":{"rendered":"<p><!-- Integration of Abstract Functions --><\/p>\n<p>\uc774 \uae00\uc5d0\uc11c\ub294 \ucd94\uc0c1\uacf5\uac04\uc5d0\uc11c\uc758 \uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p>\\(E\\)\ub97c \ub178\ub984\ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uace0, \\(K\\)\ub97c \ub2eb\ud78c\uad6c\uac04 \\([0,\\,1]\\)\uc774\ub77c\uace0 \ud558\uc790. \\(K\\)\uc5d0\uc11c \\(E\\)\ub85c\uc758 \ud568\uc218 \\(x = x(t)\\)\ub97c \uc0dd\uac01\ud558\uc790. (\uc120\ud615\uc778 \uacbd\uc6b0\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc77c\ubc18\uc801\uc778 \ud568\uc218\ub97c \uc0dd\uac01\ud558\uc790.) \uc55e\uc73c\ub85c \uc774\ub7ec\ud55c \ud568\uc218\ub97c \uad6c\uac04 \\([0,\\,1]\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c <span class=\"defined\">\ucd94\uc0c1\ud654\ub41c \ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub97c \uac83\uc774\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \ud568\uc218\uc5d0 \ub300\ud558\uc5ec, \ud574\uc11d\ud559\uc758 \uae30\ubcf8\uc801\uc778 \uc5f0\uc0b0\uc744 \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uc720\ub3c4\ud558\uc790.<\/p>\n<h3>\uc801\ubd84\uc758 \uc815\uc758<\/h3>\n<p>\ub2eb\ud78c\uad6c\uac04 \\([a,\\,b]\\)\ub97c \ub2eb\ud78c \ubd80\ubd84\uad6c\uac04 \\([t_i ,\\, t_{i+1}]\\)\ub85c \ub098\ub208 <span class=\"defined\">\ubd84\ud560<\/span> \\(A = A[t_0 ,\\, t_1 ,\\, \\cdots ,\\, t_n ]\\)\uc744 \uc0dd\uac01\ud558\uc790. \uc774\ub54c<br \/>\n\\[a = t_0 < t_1 < \\cdots < t_n = b\\]\n\uc774\ub2e4.\n\ubd84\ud560 \\(B = B[t_0 ' ,\\, t_1 ' ,\\, \\cdots ,\\, t_m ' ]\\)\uc774 \\(A = A[t_0 ,\\, t_1 ,\\, \\cdots ,\\, t_n ]\\)\uc758 <span class=\"defined\">\uc138\ub828\ubd84\ud560<\/span>\uc774\ub77c\ub294 \uac83\uc740 \ubaa8\ub4e0 \uad6c\uac04 \\([t_i &#8216; ,\\, t_{i+1} &#8216;]\\)\uc774 \uc5b4\ub5a4 \uad6c\uac04 \\([t_i ,\\, t_{i+1}]\\)\uc5d0 \ud3ec\ud568\ub428\uc744 \uc758\ubbf8\ud55c\ub2e4. \ub530\ub77c\uc11c \ubd84\ud560 \\(B\\)\uc5d0\uc11c\ub294 \\(A\\)\uc758 \ubaa8\ub4e0 \uad6c\uac04\uc774 \ub2e4\uc2dc \ubd84\ud560\ub41c\ub2e4. \ub9cc\uc57d \ubd84\ud560 \\(A\\)\uc758 \ubaa8\ub4e0 \uad6c\uac04 \\([t_i ,\\, t_{i+1}]\\)\uc758 \uae38\uc774\uac00 \uc591\uc218 \\(\\tau\\)\ub97c \ucd08\uacfc\ud558\uc9c0 \uc54a\uc73c\uba74, \uc989 \\(t_{i+1} &#8211; t_i \\le \\tau\\)\uc774\uba74, \\(A\\)\ub97c \uad6c\uac04 \\([a,\\,b]\\)\uc758 <span class=\"defined\">\\(\\tau\\)-\ubd84\ud560<\/span>\uc774\ub77c\uace0 \ud558\uace0 \\(A_\\tau\\)\ub85c \ud45c\uae30\ud55c\ub2e4.<\/p>\n<p>\\(x\\in E\\)\uc774\uace0 \\(t\\in [a,\\,b]\\)\uc778 \ucd94\uc0c1\ud654\ub41c \ud568\uc218 \\(x(t)\\)\uc5d0 \ub300\ud558\uc5ec, \ubd84\ud560 \\(A = A[t_0 ,\\, t_1 ,\\, \\cdots ,\\, t_n ]\\)\uc5d0 \ub300\ud55c \ud569<br \/>\n\\[S(A,\\,x(t)) = \\sum_{i=0}^{n-1} x(t_i )(t_{i+1} &#8211; t_i ) \\tag{1}\\]<br \/>\n\uc744 <span class=\"defined\">\uc801\ubd84\ud569<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span><br \/>\n\\(x(t)\\)\uac00 \uc644\ube44\uacf5\uac04 \\(E\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\uc774\uace0 \\(t\\in [a,\\,b]\\)\uc774\uba70, \\(A_{\\tau_n}\\)\uc774 \uad6c\uac04 \\([0,\\,1]\\)\uc758 \ubd84\ud560\uc758 \uc218\uc5f4\ub85c\uc11c \\(n \\to \\infty\\)\uc77c \ub54c \\(\\tau_n \\to 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c \uc801\ubd84\ud569 \\(S(A_{\\tau_n} ,\\, x(t))\\)\ub97c \ub9cc\ub4e4\uba74, \uc774\ub4e4\uc740 \\(n\\to\\infty\\)\uc77c \ub54c \uadf9\ud55c \\(S\\)\ub85c \uc218\ub834\ud55c\ub2e4. \uadf9\ud55c \\(S\\)\ub294 \\(A_{\\tau_n}\\)\uc758 \uc120\ud0dd\uc5d0 \uc758\uc874\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<\/div>\n<p>\uc815\ub9ac\uc758 \uc99d\uba85\uc740 \ub2e4\uc74c \uc138 \ubcf4\uc870\uc815\ub9ac\uc5d0 \uae30\ucd08\ud55c\ub2e4.<\/p>\n<div class=\"box\">\n<p><span class=\"lemma\">\ubcf4\uc870\uc815\ub9ac 1.<\/span><br \/>\n\\(t\\in [a,\\,b]\\)\uc778 \ubaa8\ub4e0 \uc5f0\uc18d\ud568\uc218 \\(x(t)\\in E\\)\ub294 \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \uc989, \ubaa8\ub4e0 \uc591\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec, \uc591\uc218 \\(\\tau\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \\(t &#8216; ,\\) \\(t &#8216; &#8216; \\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert t &#8216; &#8211; t &#8216; &#8216; \\rvert < \\tau\\)\uc77c \ub54c\ub9c8\ub2e4\n\\[\\lVert x(t ' ) - x(t ' ' ) \\rVert < \\epsilon \\tag{2}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(\\varphi (t,\\, t &#8216; )\\)\ub97c \uc815\uc0ac\uac01\ud615 \\(a \\le t \\le b,\\) \\(a \\le t &#8216; \\le b\\) \uc704\uc5d0\uc11c<br \/>\n\\[\\varphi (t,\\, t &#8216; ) = \\lVert x(t) &#8211; x(t &#8216; ) \\rVert\\]<br \/>\n\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud558\uc790.<br \/>\n\ud568\uc218 \\(\\varphi\\)\ub294 \uc774 \uc815\uc0ac\uac01\ud615 \uc704\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c, \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \ub530\ub77c\uc11c \ubaa8\ub4e0 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\lvert \\varphi ( t,\\, t &#8216; &#8216; ) &#8211; \\varphi ( t &#8216; ,\\, t &#8216; &#8216; ) \\rvert < \\epsilon \\,\\, \\text{for} \\,\\, \\lvert t - t ' \\rvert < \\tau \\tag{3}\\]\n\uac00 \uc131\ub9bd\ud558\ub3c4\ub85d \ud558\ub294 \uc218 \\(\\tau\\)\ub97c \uc815\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\ub610\ud55c, \\(t &#8216; = t\\)\uc77c \ub54c \\(\\varphi (t,\\,t) = \\lVert x(t) &#8211; x(t) \\rVert =0\\)\uc774\ub2e4. \ub530\ub77c\uc11c (3)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\varphi (t,\\,t &#8216; ) = \\lvert \\varphi (t,\\,t &#8216; ) &#8211; \\varphi (t ,\\,t) \\rvert < \\epsilon \\,\\, \\text{for} \\,\\,  \\lvert t-t ' \\rvert < \\tau\\]\n\ub610\ub294\n\\[\\lVert x(t ' ) -x(t) \\rVert < \\epsilon \\,\\, \\text{for} \\,\\,  \\lvert t-t ' \\rvert < \\tau . \\tag*{\\(\\blacksquare\\)}\\]\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"lemma\">\ubcf4\uc870\uc815\ub9ac 2.<\/span><br \/>\n\ub2eb\ud78c\uad6c\uac04 \\([a,\\,b]\\)\uc758 \ubd84\ud560 \\(W\\)\uac00 \\(\\tau\\)-\ubd84\ud560 \\(V = V_\\tau\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\uba74<br \/>\n\\[\\lVert S(V,\\,x(t)) &#8211; S(W,\\,x(t)) \\rVert \\le \\epsilon (b-a)\\tag{4}\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\uc810 \\(t &#8216; \\)\uc774 \\(\\tau\\)-\ubd84\ud560 \\(V_\\tau\\)\uc758 \uad6c\uac04 \\([t_i ,\\, t_{i+1}]\\)\uc5d0 \uc18d\ud558\uba74<br \/>\n\\[\\lvert t &#8216; &#8211; t_i \\rvert \\le t_{i+1} &#8211; t_i \\le \\tau\\]<br \/>\n\uc774\ub2e4.<br \/>\n\ub530\ub77c\uc11c (2)\uc5d0 \uc758\ud558\uc5ec,<br \/>\n\\[\\lVert x(t &#8216; ) &#8211; x(t_i ) \\rVert \\le \\epsilon .\\tag{5}\\]<br \/>\n\\(n,\\) \\(m\\)\uc744 \uac01\uac01 \\(V,\\) \\(W\\)\uc758 \uad6c\uac04 \\([t_i ,\\, t_{i+1}],\\) \\([t_j &#8216; ,\\, t_{j+1} &#8216; ]\\)\uc758 \uac1c\uc218\ub77c\uace0 \ud558\uc790. \\(W\\)\uac00 \\(V\\)\uc758 \uc138\ub828\ubd84\ud560\uc774\ubbc0\ub85c, \uac01 \uc810 \\(t_\\ell ,\\) \\(\\ell = 1,\\) \\(2,\\) \\(\\cdots,\\) \\(n\\)\uc740 \uc810 \\(t_j &#8216;\\) \uc911 \ud558\ub098\uc640 \uc77c\uce58\ud55c\ub2e4. \uc989 \\(t_{k_{\\ell}} &#8216; = t_{\\ell} \\)\uc774\ub77c\uace0 \ud558\uc790. \ub530\ub77c\uc11c \ub2e4\uc74c \uc21c\uc11c\ub97c \uac16\ub294\ub2e4.<br \/>\n\\[0 = k_0 < k_1 < \\cdots < k_n = m ,\\,\\, m \\ge n.\\]\n\ub530\ub77c\uc11c \\(V\\)\uc758 \uad6c\uac04 \\([t_\\ell ,\\, t_{\\ell +1}]\\)\uc740 \ubd84\ud560 \\(W\\)\uc758 \\(k_{\\ell +1} - k_{\\ell}\\)\uac1c\uc758 \uad6c\uac04 \\([t_j ' ,\\, t_{j+1} ']\\)\ub85c \ubd84\ud560\ub41c\ub2e4. \ub2e8, \uc5ec\uae30\uc11c \\(j = k_{\\ell} ,\\) \\(k_\\ell +1 ,\\) \\(\\cdots ,\\) \\(k_{\\ell +1} -1\\)\uc774\ub2e4. (5)\uc5d0 \uc758\ud558\uc5ec, \ubaa8\ub4e0 \\(j\\)\uc5d0 \ub300\ud574\n\\[\\lVert x(t_j ' ) - x(t_i ) \\rVert \\le \\epsilon\\]\n\uc774\ub2e4.\n\ub530\ub77c\uc11c\n\\[\\begin{align}\nS(V,\\,x(t)) &#038;= \\sum_{\\ell=0}^{n-1} x(t_\\ell )(t_{\\ell +1} - t_\\ell ) \\\\[6pt]\n&#038;= \\sum_{\\ell =0}^{n-1} x(t_\\ell ) \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} (t_{j+1} ' - t_j ' ),\\\\[8pt]\nS(W,\\,x(t)) &#038;= \\sum_{j=0}^{m-1} x(t_j ' )(t_{j+1} ' - t_j ' ) \\\\[6pt]\n&#038;= \\sum_{\\ell =0}^{n-1} \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} x(t_j ' ) (t_{j+1} ' - t_j ' ),\\\\[8pt]\n\\lVert S(V,\\,x(t)) -  S(W,\\,x(t)) \\rVert \n&#038;=\\left\\lVert \\sum_{\\ell =0}^{n-1} \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} [x(t_\\ell ) - x(t_j ' )] (t_{j+1} ' - t_j ' ) \\right\\rVert \\\\[6pt]\n&#038;\\le \\sum_{\\ell =0}^{n-1} \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} \\lVert x(t_\\ell ) - x(t_j ' ) \\rVert (t_{j+1} ' - t_j ' ) \\\\[6pt]\n&#038;\\le \\sum_{\\ell =0}^{n-1} \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} \\epsilon (t_{j+1} ' - t_j ' ) \\\\[6pt]\n&#038;= \\epsilon (b-a) .\\tag*{\\(\\blacksquare\\)}\n\\end{align}\\]\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"lemma\">\ubcf4\uc870\uc815\ub9ac 3.<\/span><br \/>\n\\(V_\\tau\\)\uc640 \\(V_{\\tau &#8216;}\\)\uc744 \uac01\uac01 \ub2eb\ud78c\uad6c\uac04 \\([a,\\,b]\\)\uc758 \uc784\uc758\uc758 \\(\\tau ,\\) \\(\\tau &#8216; \\) \ubd84\ud560\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[\\lVert S(V_\\tau ,\\, x(t)) &#8211; S(V_{\\tau &#8216;} ,\\, x(t)) \\rVert \\le (\\epsilon_\\tau + \\epsilon_{\\tau &#8216;})(b-a) .\\tag{6}\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(V_\\tau\\)\uc640 \\(V_{\\tau &#8216;}\\) \ub458 \ub2e4\uc758 \uc138\ub828\ubd84\ud560\uc778 \uad6c\uac04 \\([a,\\,b]\\)\uc758 \ubd84\ud560 \\(W\\)\ub97c \ud56d\uc0c1 \uc120\ud0dd\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\uba74 (4)\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\lVert S(V_\\tau ,\\, x(t)) &#8211; S(W,\\, x(t)) \\rVert \\le \\epsilon_\\tau (b-a),\\\\[6pt]<br \/>\n\\lVert S(V_{\\tau &#8216;} ,\\, x(t)) &#8211; S(W,\\, x(t)) \\rVert \\le \\epsilon_{\\tau &#8216;} (b-a)\\]<br \/>\n\uc774\uace0, \uc5ec\uae30\uc11c<br \/>\n\\[\\lVert S(V_\\tau ,\\, x(t)) &#8211; S(V_{\\tau &#8216;} ,\\, x(t)) \\rVert \\le (\\epsilon_\\tau + \\epsilon_{\\tau &#8216;} ) (b-a)\\tag*{\\(\\blacksquare\\)}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.\n<\/p>\n<\/div>\n<p>\uc774\uc81c \uc815\ub9ac 1\uc744 \uc99d\uba85\ud558\uc790. \uc784\uc758\ub85c \uc8fc\uc5b4\uc9c4 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec, \uad6c\uac04 \\([a,\\,b]\\)\uc758 \\(\\tau_n\\)-\ubd84\ud560\uc758 \uc218\uc5f4 \\(V_{\\tau_n}\\)\uc744 \uc0dd\uac01\ud558\uc790. \uc5ec\uae30\uc11c \\(n\\to\\infty\\)\uc77c \ub54c \\(\\tau_n \\to 0\\)\uc774\ub2e4. \ubcf4\uc870\uc815\ub9ac 3\uc5d0 \uc758\ud558\uc5ec, \ub300\uc751\ud558\ub294 \uc801\ubd84\ud569 \\(S(V_{\\tau_n} ,\\, x(t))\\)\uc5d0 \ub300\ud558\uc5ec \ubd80\ub4f1\uc2dd<br \/>\n\\[\\lVert S(V_{\\tau_n} ,\\, x(t)) &#8211; S(V_{\\tau_{n+1}} ,\\, x(t)) \\lVert \\le (\\epsilon_{\\tau_n} + \\epsilon_{\\tau_{n+1}} )(b-a)\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<p>\uc774\uac83\uc774 \ubaa8\ub4e0 \\(\\epsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud558\ubbc0\ub85c, \uc218\uc5f4 \\(S(V_{\\tau_n},\\, x(t))\\)\ub294 \ucf54\uc2dc \uc218\uc5f4\uc774\uace0, \\(S(V_{\\tau_n} ,\\, x(t)) \\in E\\)\uc774\uba70 \\(E\\)\uac00 \uc644\ube44\uc774\ubbc0\ub85c, \uc774 \uc218\uc5f4\uc740 \\(E\\)\uc5d0\uc11c \uadf9\ud55c \\(S\\)\ub97c \uac16\ub294\ub2e4.<\/p>\n<p>\uc774\uc81c \\(V_{\\tau_n &#8216;}\\)\uc744 \uad6c\uac04 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc758 \ub610 \ub2e4\ub978 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(\\tau_n &#8216; \\to 0\\)\uc774\ub2e4. \uc55e\uc5d0\uc11c \ubc1d\ud78c \uacb0\uacfc\ub97c \uc0ac\uc6a9\ud558\uba74, \\(S(V_{\\tau_n &#8216;},\\,x(t))\\)\ub294 \\(n\\to\\infty\\)\uc77c \ub54c \uc801\ub2f9\ud55c \uac12 \\(S_1\\)\uc73c\ub85c \uc218\ub834\ud55c\ub2e4. \uc774\uc81c \\(S = S_1\\)\uc784\uc744 \uc99d\uba85\ud558\uba74 \ub41c\ub2e4.<\/p>\n<p>\uc774\ub97c \uc704\ud558\uc5ec, \uc218\uc5f4 \\(V_{\\tau_1} ,\\) \\(V_{\\tau_1 &#8216;},\\) \\(V_{\\tau_2} ,\\) \\(V_{\\tau_2 &#8216;} ,\\) \\(\\cdots\\)\uc744 \uc0dd\uac01\ud558\uc790. \uc774\ub4e4 \uc218\uc5f4\uc5d0 \ub300\uc751\ud558\ub294 \uc801\ubd84\ud569 \\(S(V_{\\tau_n} ,\\, x(t)),\\) \\(S(V_{\\tau_n &#8216;} ,\\, x(t))\\)\ub294 \uac01\uac01 \uc218\ub834\ud558\ub294 \uc218\uc5f4\uc744 \uc774\ub8ec\ub2e4. \uc774\ub4e4\uc758 \uadf9\ud55c\uc774 \\(S_2\\)\uc640 \uac19\uc73c\uba74, \uc774\ub4e4\uc740 \uadf9\ud55c \\(S\\)\uc640 \\(S_1\\)\uacfc \uac19\uc544\uc57c \ud55c\ub2e4. \ub530\ub77c\uc11c \\(S = S_1 = S_2\\)\uc774\uace0, \uc815\ub9ac\uac00 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n<p>\\(S\\)\ub97c \\([a,\\,b]\\)\uc5d0\uc11c \\(x(t)\\)\uc758 <span class=\"defined\">\uc801\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uace0<br \/>\n\\[\\int_a^b x(t) dt\\]<br \/>\n\ub85c \ud45c\uae30\ud55c\ub2e4.\n<\/p>\n<h3>\uc801\ubd84\uc758 \uc131\uc9c8<\/h3>\n<p>\ub2e4\uc74c \uc131\uc9c8\uc740 \uc801\ubd84\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uace7\ubc14\ub85c \uc720\ub3c4\ub418\ub294 \uac83\ub4e4\uc774\ub2e4.<\/p>\n<p>[1] \uc801\ubd84\uc740 \uac00\ubc95\uc801\uc774\ub2e4.<br \/>\n\\[\\int_a^b [x(t) + y(t)] dt = \\int_a^b x(t) dt + \\int_a^b y(t)dt.\\]<br \/>\n[2] \\(\\lambda\\)\uac00 \uace0\uc815\ub41c \uc2a4\uce7c\ub77c\uc774\uba74<br \/>\n\\[\\int_a^b \\lambda x(t) dt = \\lambda \\int_a^b x(t) dt.\\]<br \/>\n[3] \ub610\ud55c<br \/>\n\\[\\left\\lVert \\int_a^b x(t) dt \\right\\rVert \\le \\int_a^b \\lVert x(t) \\rVert dt \\tag{7}\\]<br \/>\n\uc774\ub2e4.<br \/>\n\uc65c\ub0d0\ud558\uba74,<br \/>\n\\[\\begin{align}<br \/>\n\\lVert S(V_\\tau ,\\, x(t)) \\rVert<br \/>\n&#038;= \\left\\lVert \\sum_{i=0}^{n-1} x(t_i )(t_{i+1} &#8211; t_i ) \\right\\rVert \\\\[6pt]<br \/>\n&#038;\\le \\sum_{i=0}^{n-1} \\lVert x(t) \\rVert (t_{i+1} &#8211; t_i ) \\\\[6pt]<br \/>\n&#038;= S(V_\\tau ,\\, \\lVert x(t) \\rVert ) \\tag{8}<br \/>\n\\end{align}\\]<br \/>\n\uc774\uae30 \ub54c\ubb38\uc774\ub2e4.<br \/>\n\\(\\tau \\to 0\\)\uc77c \ub54c, \uc801\ubd84\ud569\uc740 \uc801\ubd84<br \/>\n\\[\\int_a^b x(t) dt \\,\\,\\,\\text{\ubc0f}\\,\\,\\, \\int_a^b \\lVert x(t) \\rVert dt\\]<br \/>\n\ub85c \uc218\ub834\ud558\uace0, \ubd80\ub4f1\uc2dd (8)\uc758 \uc591\ubcc0\uc740 \ubd80\ub4f1\uc2dd (7)\uc758 \uc591\ubcc0\uc73c\ub85c \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>[4] \uc6d0\uc18c \\(x\\in E\\)\uc5d0 \ub300\ud558\uc5ec \uc6d0\uc18c \\(y\\in E_1\\)\uacfc\uc758 \uc6b0\uce21 \uacf1\uc148(\ub610\ub294 \uc88c\uce21 \uacf1\uc148)\uc774 \uc815\uc758\ub418\uc5b4 \uc788\uc73c\uba74,<br \/>\n\\[\\int_a^b x(t) y dt = \\int_a^b x(t) dt \\,y.\\tag{9}\\]<br \/>\n\uc65c\ub0d0\ud558\uba74,<br \/>\n\\[\\begin{align}<br \/>\nS(V_\\tau ,\\, x(t)y )<br \/>\n&#038;= \\sum_{i=0}^{n-1} x(t_i ) y(t_{i+1} &#8211; t_i ) \\\\[6pt]<br \/>\n&#038;=\\left(\\sum_{i=0}^{n-1} x(t_i )(t_{i+1} &#8211; t_i ) \\right) y \\\\[6pt]<br \/>\n&#038;= S(V_\\tau ,\\, x(t)) y \\tag{10}<br \/>\n\\end{align}\\]<br \/>\n\uac00 \\(\\tau\\)-\ubd84\ud560 \\(V_\\tau (t_0 ,\\, t_1 ,\\, \\cdots ,\\, t_n )\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud558\uae30 \ub54c\ubb38\uc774\ub2e4. \\(\\tau \\to 0\\)\uc77c \ub54c, \ub4f1\uc2dd (10)\uc758 \uc591\ubcc0\uc740 (9)\uc758 \uc591\ubcc0\uc73c\ub85c \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\ub9c8\ucc2c\uac00\uc9c0\ub85c, \uc88c\uce21 \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec,<br \/>\n\\[\\int_a^b yx(t)dt = y\\int_a^b x(t)dt \\tag{9\\( &#8216; \\)}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\\(A\\)\uac00 \\(E\\)\uc5d0\uc11c \\(E_1\\)\ub85c\uc758 \uc120\ud615\uc5f0\uc0b0\uc790\uc774\uace0, \\(x(t)\\in E\\)\uc774\uba74,<br \/>\n\\[\\int_a^b Ax(t)dt = A \\int_a^b x(t)dt\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><br \/>\n\\(A = A(t)\\)\uac00 \\(t\\)\uc5d0 \ub300\ud558\uc5ec \uc5f0\uc18d\uc774\uace0 \\((E \\to E_1 )\\)\uc778 \uc5f0\uc0b0\uc790\uc774\uba70, \\(x\\in E\\)\uc774\uba74,<br \/>\n\\[\\int_a^b A(t) x dt = \\left( \\int_a^b A(t) dt \\right) x\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uccab \ubc88\uc9f8 \uc608\uc5d0\uc11c \uc5f0\uc0b0\uc790 \\(A\\)\ub294 \uc0c1\uc218 \uc88c\uce21 \uc778\uc218\uc774\uace0, \ub450 \ubc88\uc9f8 \uc608\uc5d0\uc11c \\(x\\)\ub294 \uc0c1\uc218 \uc6b0\uce21 \uc778\uc218\uc774\ub2e4.<\/p>\n<p>\ud2b9\ud788, \\(f\\)\uac00 \\(\\overline{E}\\)\uc758 \uc120\ud615\ubc94\ud568\uc218\uc774\uba74,<br \/>\n\\[f\\left( \\int_a^b x(t) dt \\right) = \\int_a^b f(x(t)) dt,\\\\[6pt]<br \/>\n\\int_a^b f(t) (x) dt = \\left( \\int_a^b f(t) dt \\right) (x).\\tag{10\\( &#8216; \\)}\\]\n<\/p>\n<p>[5] \ud568\uc218 \\(x=x(t),\\) \\(x\\in E,\\) \\(t\\in [a,\\,b]\\)\uac00 \\(t\\)\uc5d0 \uad00\ud55c \uc5f0\uc18d\uc778 \ub3c4\ud568\uc218 \\(x &#8216; (t) = \\frac{d}{dt} x(t)\\)\ub97c \uac00\uc9c0\uba74,<br \/>\n\\[\\int_a^b x &#8216; (t) dt = x(b) &#8211; x(a) .\\tag{11}\\]<br \/>\n\uc65c\ub0d0\ud558\uba74, (10&#8217;)\uc5d0 \uc758\ud558\uc5ec, \uc784\uc758\uc758 \uc120\ud615\ubc94\ud568\uc218 \\(L\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[L \\left( \\int_a^b x &#8216; (t) dt \\right) = \\int_a^b L [x &#8216; (t)] dt = \\int_a^b \\frac{d}{dt} \\left\\{ L[x(t)] \\right\\} dt\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uae30 \ub54c\ubb38\uc774\ub2e4.<br \/>\n\uadf8\ub7f0\ub370 \\(L[x(t)]\\)\ub294 \uce58\uc5ed\uc774 \uc218\ub85c \uc774\ub8e8\uc5b4\uc9c4 \\(t\\)\uc758 \ud568\uc218\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\int_a^b \\left\\{ L [x(t)] \\right\\} &#8216; dt = L[x(b)] &#8211; L[x(a)]\\]<br \/>\n\uc774\ub2e4.<br \/>\n\ub530\ub77c\uc11c,<br \/>\n\\[L \\left( \\int_a^b x &#8216; (t) dt \\right) = L [x(b) &#8211; x(a)].\\tag{12}\\]<br \/>\n\ub4f1\uc2dd (12)\ub294 \uc784\uc758\uc758 \uc120\ud615\ubc94\ud568\uc218 \\(L \\in \\overline{E}\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4. \uc774\ub85c\uc368 (11)\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.<\/p>\n<h3>\ubbf8\ubd84\ubc29\uc815\uc2dd\uc5d0\uc758 \uc751\uc6a9<\/h3>\n<p>\ubbf8\ubd84\ubc29\uc815\uc2dd<br \/>\n\\[\\frac{dx}{dt} = f(t,\\,x)\\tag{13}\\]<br \/>\n\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\(x = x(t)\\)\uc640 \\(f(t,\\,x)\\)\ub294 \ub178\ub984\uacf5\uac04 \\(E\\)\uc758 \uc6d0\uc18c\uc774\ub2e4. \\(t\\in [a,\\,b]\\)\ub77c\uace0 \ud558\uc790. \\(f(t,\\,x)\\)\uac00 \\(t\\)\uc5d0 \ub300\ud558\uc5ec \uc5f0\uc18d\uc774\uace0, \\(x\\)\uc5d0 \uad00\ud558\uc5ec \ub9bd\uc2dc\uce20 \uc870\uac74<br \/>\n\\[\\lVert f(t,\\,x_1 ) &#8211; f(t,\\,x_2 ) \\lVert \\le M \\lVert x_1 &#8211; x_2 \\rVert \\tag{14}\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4.<\/p>\n<p>\\(C^E [a,\\,b]\\)\ub97c \\(t\\in [a,\\,b]\\)\uc774\uace0 \\(x(t)\\in E\\)\uc778 \uc5f0\uc18d\ud568\uc218 \\(x(t)\\)\uc758 \uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \\(C^E [a,\\,b]\\)\uc5d0 \ub178\ub984\uc744<br \/>\n\\[\\lVert x \\rVert_C = \\max_{a\\le t\\le b} \\lVert x(t) \\rVert\\]<br \/>\n\ub85c \ub3c4\uc785\ud55c\ub2e4.<br \/>\n\\(E\\) \uc678\uc5d0\ub3c4, \\(C^E [a,\\,b]\\)\ub294 \uc644\ube44\uacf5\uac04\uc774\ub2e4. \uc774 \uc131\uc9c8\uc758 \uc99d\uba85\uc740 \ud2b9\uc218\ud55c \uacbd\uc6b0\uc778 \\(E=\\mathbb{R}\\)\uc774\uace0 \\(C^E [a,\\,b] = C[0,\\,1]\\)\uc778 \uacbd\uc6b0\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud55c\ub2e4.<\/p>\n<p>\ub4f1\uc2dd (13) \uc678\uc5d0\ub3c4,<br \/>\n\\[x(t) = x_0 + \\int_{t_0}^t f(t,\\,x(t)) dt,\\,\\, a\\le t_0 \\le t \\le t_0 + \\delta \\le b \\tag{15}\\]<br \/>\n\uc640 \uac19\uc740 \ub4f1\uc2dd\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<br \/>\n\uc774 \ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc744 \\(A(x)\\)\ub85c \ud45c\uae30\ud558\uc790. \\(A\\)\ub294 \\(x=x(t)\\in C^E [t_0 ,\\, t_0 + \\delta ]\\)\ub97c \uac19\uc740 \uacf5\uac04\uc758 \uc6d0\uc18c\ub85c \ubcc0\ud658\ud558\ub294 \uc5f0\uc0b0\uc790\uc774\ub2e4. \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\lVert A(x) &#8211; A(y) \\rVert_C<br \/>\n&#038;= \\left\\lVert \\int_{t_0}^t [f(t,\\,x(t)) &#8211; f(t,\\,y(t))]dt \\right\\rVert_C \\\\[6pt]<br \/>\n&#038;\\le \\int_{t_0}^{t_0 + \\delta} \\lVert f(t,\\,x(t)) &#8211; f(t,\\,y(t)) \\rVert dt.<br \/>\n\\end{align}\\]<br \/>\n(14)\uc5d0 \uc758\ud558\uc5ec,<br \/>\n\\[\\begin{align}<br \/>\n\\lVert A(x) &#8211; A(y) \\rVert_C &#038;\\le M\\int_{t_0}^{t_0 + \\delta} \\lVert x(t) &#8211; y(t) \\rVert dt\\\\[6pt]<br \/>\n&#038;\\le M \\delta \\max_{t_0 \\le t \\le t_0 + \\delta} \\lVert x(t) &#8211; y(t) \\rVert \\\\[6pt]<br \/>\n&#038;=M \\delta \\lVert x(t) &#8211; y(t) \\rVert_C \\tag{16}<br \/>\n\\end{align}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<br \/>\n\ub9cc\uc57d \\(M\\delta < 1\\)\uc774\uba74, (16)\uc5d0 \uc758\ud558\uc5ec \uc5f0\uc0b0\uc790 \\(A\\)\ub294 \uacf5\uac04 \\(C^E [a,\\,b]\\)\ub97c \uac19\uc740 \uacf5\uac04\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ucd95\uc18c\uc0ac\uc0c1\uc774\uba70, \ub530\ub77c\uc11c (15)\uc758 \ud574\uac00 \uc815\ud655\ud788 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<p>\ub4f1\uc2dd (15)\ub294 \ucd08\uae43\uac12 \\(x(t_0 ) = x_0\\)\uc5d0 \ub300\ud558\uc5ec (13)\uacfc \ub3d9\uce58\uc774\ub2e4. \ub530\ub77c\uc11c (13)\uc740 \uad6c\uac04 \\([t_0 ,\\, t_0 + \\delta ]\\)\uc5d0\uc11c \uc815\ud655\ud788 \ud558\ub098\uc758 \ud574\ub97c \uac16\ub294\ub2e4.<\/p>\n<p>\ud2b9\ud788, \uc774 \ubc29\uc815\uc2dd\uc740 \uc784\uc758\uc758 \ucd08\uae43\uac12 \\(x(a) = x_0\\)\uc5d0 \ub300\ud558\uc5ec \uad6c\uac04 \\([a,\\,a+\\delta ]\\)\uc5d0\uc11c \uc815\ud655\ud788 \ud558\ub098\uc758 \ud574 \\(x(t)\\)\ub97c \uac16\ub294\ub2e4. \\(x(t)\\)\ub294 \uc804\uccb4 \uad6c\uac04 \\([a,\\,b]\\)\ub85c \ud655\uc7a5\ub420 \uc218 \uc788\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \\(a+\\delta < b\\)\uc774\uace0 \\(x(a+\\delta ) = x_1\\)\uc774\uba74, \ucd08\uae43\uac12 \\(x_1\\)\uc744 \uac16\ub294 \uad6c\uac04 \\([a+\\delta ,\\, a+2 \\delta ]\\)\uc5d0\uc11c\uc758 \ud574\ub97c \uc774 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec \uad6c\uc131\ud560 \uc218 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><br \/>\n\ub9cc\uc57d \\(E\\)\uac00 \\(n\\)\ucc28\uc6d0 \uacf5\uac04\uc774\uba74, \\(n\\)\uac1c\uc758 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc758 \uacc4\uc5d0 \ub300\ud55c \uc798 \uc54c\ub824\uc9c4 \uc874\uc7ac\uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><br \/>\n\ub9cc\uc57d \\(E\\)\uac00 \uacf5\uac04 \\(l_p ,\\) \\(c,\\) \\(m\\) \ub4f1 \uc911 \ud558\ub098\uc774\uba74, \ubb34\ud55c \ubbf8\ubd84\ubc29\uc815\uc2dd\uacc4\uc758 \ub300\uc751\ud558\ub294 \uc885\ub958\uc758 \ud574\uc5d0 \ub300\ud55c \uc874\uc7ac\uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.<\/span><br \/>\n\uc801\ubd84-\ubbf8\ubd84\ubc29\uc815\uc2dd<br \/>\n\\[\\frac{dy}{dt} = f(t,\\,y) + \\int_a^b K(t,\\,s) y(s) ds \\tag{17}\\]<br \/>\n\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\(f(t,\\,y)\\)\uac00 \ub9bd\uc2dc\uce20 \uc870\uac74 (14)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uace0, \ud575 \\(K(t,\\,s)\\)\uac00 \uc720\uacc4, \uc989 \\(\\lvert K(t,\\,s) \\rvert < M_1\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>(17)\uc758 \uc6b0\ubcc0\uc744 \\(F(t,\\,y)\\)\ub85c \ud45c\uae30\ud558\uba74 \ubd80\ub4f1\uc2dd<br \/>\n\\[\\begin{align}<br \/>\n\\lvert F(t,\\,y) &#8211; F(t,\\,z) \\rvert<br \/>\n&#038;\\le M \\lvert y-z \\rvert + M_1 (b-a) \\lvert y-z \\rvert \\\\[6pt]<br \/>\n&#038;= (M+M_1 (b-a)) \\lvert y-z \\rvert<br \/>\n\\end{align}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.<br \/>\n\\(F(t,\\,y)\\)\ub294 \ub9bd\uc2dc\uce20 \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub530\ub77c\uc11c (17)\uc740 \uc784\uc758\uc758 \ucd08\uae43\uac12 \\(y=y_0\\)\uc5d0 \ub300\ud558\uc5ec \uc720\uc77c\ud55c \ud574\ub97c \uac16\ub294\ub2e4.\n<\/p>\n<\/div>\n<h3>\ucc38\uace0\ubb38\ud5cc<\/h3>\n<ul class=\"bracket\">\n<li>\ub958\uc2a4\ud14c\ub974\ub2c8\ud06c(L. A. Liusternik)\uc640 \uc18c\ubcfc\ub808\ud504(V. J. Sobolev), \u300e\ud568\uc218\ud574\uc11d\ud559(Elements of Functional Analysis)\u300f (\uc564\uc11c\ub2c8 \ub77c\ubc14\ub808(Anthony E. Labarre) \ubc88\uc5ed), \ud504\ub808\ub354\ub9ad \uc6c5\uac00\ub974 \ucd9c\ud310\uc0ac(Frederick Ungar Publishing Company, New York), 173-178\ucabd.\n<\/li>\n<\/ul>\n<p><!--\n\n\n\n<p>In this post we will study integration in abstract spaces.<\/p>\n\n\n\n\n\n<p>Let \\(E\\) be a normed linear space and \\(K\\) the closed interval \\([a,\\,b]\\) of the real number line. We consider an operator \\(x = x(t),\\) which need not be linear and maps \\(K\\) into \\(E.\\) In the following, we will call such an operator an <span class=\"defined\">abstract function<\/span> on the interval \\([a,\\,b].\\)<\/p>\n\n\n\n\n\n<p>For these functions, we shall define and deduce properties of the fundamental operations of analysis.<\/p>\n\n\n\n\n\n<h3>Definition of the Integral<\/h3>\n\n\n\n\n\n<p>We consider a <span class=\"defined\">partition<\/span> \\(A = A[t_0 ,\\, t_1 ,\\, \\cdots ,\\, t_n ]\\) of the closed interval \\([a,\\,b]\\) into closed subintervals \\([t_i ,\\, t_{i+1}]\\) with\n\\[a = t_0 < t_1 < \\cdots < t_n = b.\\]\nThe partition \\(B = B[t_0 ' ,\\, t_1 ' ,\\, \\cdots ,\\, t_m ' ]\\) is called a <span class=\"defined\">refinement<\/span> of \\(A = A[t_0 ,\\, t_1 ,\\, \\cdots ,\\, t_n ],\\) if every interval \\([t_i ' ,\\, t_{i+1} ']\\) is contained in an interval \\([t_i ,\\, t_{i+1}].\\) In the partition \\(B,\\) therefore, every interval of \\(A\\) is again partitioned. If every interval \\([t_i ,\\, t_{i+1}]\\) of the partition \\(A\\) has a length which does not exceed a positive number \\(\\tau ,\\) \\(t_{i+1} - t_i \\le \\tau ,\\) then \\(A\\) is called <span class=\"defined\">\\(\\tau\\)-partition<\/span> of the interval \\([a,\\,b]\\) and is designated by \\(A_\\tau .\\)<\/p>\n\n\n\n\n\n<p>We call\n\\[S(A,\\,x(t)) = \\sum_{i=0}^{n-1} x(t_i )(t_{i+1} - t_i ) \\tag{1}\\]\nan <span class=\"defined\">integral sum<\/span> of the abstract function \\(x(t)\\) with \\(x\\in E\\) and \\(t\\in [a,\\,b]\\) for the partition \\(A = A[t_0 ,\\, t_1 ,\\, \\cdots ,\\, t_n ].\\)<\/p>\n\n\n\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">Theorem 1.<\/span>\nLet \\(x(t)\\) be a continuous function in a complete space \\(E\\) with \\(t\\in [a,\\,b]\\) and \\(A_{r_n}\\) a sequence of partitions of the interval \\([0,\\,1]\\) such that \\(\\tau_n \\to 0\\) as \\(n \\to \\infty .\\) If we now form the integral sums \\(S(A_{\\tau_n} ,\\, x(t)),\\) then they tend to a limit \\(S\\) as \\(n\\to\\infty .\\) The limit \\(S\\) does not depend on the choice of the \\(A_{\\tau_n} .\\)<\/p>\n\n\n<\/div>\n\n\n\n\n\n<p>The proof of the theorem is based on the following three lemmas.<\/p>\n\n\n\n\n\n<div class=\"box\">\n\n\n<p><span class=\"lemma\">Lemma 1.<\/span>\nEvery continuous function \\(x(t)\\in E\\) with \\(t\\in [a,\\,b]\\) is uniformly continuous, i.e., for every positive number \\(\\epsilon\\) we can determine a number \\(\\tau\\) such that for \\(t ' ,\\) \\(t ' ' \\in [a,\\,b],\\)\n\\[\\lVert x(t ' ) - x(t ' ' ) \\rVert < \\epsilon , \\tag{2}\\]\nwhenever \\(\\lvert t ' - t ' ' \\rvert < \\tau.\\)\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof-eng\">\n\n\n<p class=\"proofname\">Proof<\/p>\n\n\n\n\n<p>We use \\(\\varphi (t,\\, t ' )\\) to designate the function which is defined on the square \\(a \\le t \\le b,\\) \\(a \\le t ' \\le b,\\) by\n\\[\\varphi (t,\\, t ' ) = \\lVert x(t) - x(t ' ) \\rVert .\\]\nThe function \\(\\varphi\\) is continuous on this square, therefore uniformly continuous. Consequently, we can determine for every \\(\\epsilon > 0,\\) a number \\(\\tau\\) such that\n\\[\\lvert \\varphi ( t,\\, t ' ' ) - \\varphi ( t ' ,\\, t ' ' ) \\rvert < \\epsilon \\,\\, \\text{for} \\,\\, \\lvert t - t ' \\rvert < \\tau \\tag{3}\\]\nis fulfilled.<\/p>\n\n\n\n\n<p>Furthermore, for \\(t ' = t,\\) \\(\\varphi (t,\\,t) = \\lVert x(t) - x(t) \\rVert =0.\\) Hence according to (3)\n\\[\\varphi (t,\\,t ' ) = \\lvert \\varphi (t,\\,t ' ) - \\varphi (t ,\\,t) \\rvert < \\epsilon \\,\\, \\text{for} \\,\\,  \\lvert t-t ' \\rvert < \\tau\\]\nor\n\\[\\lVert x(t ' ) -x(t) \\rVert < \\epsilon \\,\\, \\text{for} \\,\\,  \\lvert t-t ' \\rvert < \\tau . \\tag*{\\(\\blacksquare\\)}\\]\n<\/p>\n\n\n<\/div>\n\n\n\n\n\n\n\n<div class=\"box\">\n\n\n<p><span class=\"lemma\">Lemma 2.<\/span>\nIf the partition \\(W\\) of the closed interval \\([a,\\,b]\\) is a refinement of the \\(\\tau\\)-partition \\(V = V_\\tau\\) then\n\\[\\lVert S(V,\\,x(t) - S(W,\\,x(t)) \\rVert \\le \\epsilon (b-a).\\tag{4}\\]\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof-eng\">\n\n\n<p class=\"proofname\">Proof<\/p>\n\n\n\n\n<p>If the point \\(t ' \\) belongs to the interval \\([t_i ,\\, t_{i+1}]\\) of the \\(\\tau\\)-partition \\(V_\\tau ,\\) then\n\\[\\lvert t ' - t_i \\rvert \\le t_{i+1} - t_i \\le \\tau .\\]\nHence, on account of (2),\n\\[\\lVert x(t ' ) - x(t_i ) \\rVert \\le \\epsilon .\\tag{5}\\]\nLet \\(n,\\) \\(m\\) be the number of intervals \\([t_i ,\\, t_{i+1}],\\) \\(t_j ' ,\\, t_{j+1} ' ]\\) of \\(V,\\) \\(W,\\) respectively. Since \\(W\\) is a refinement of \\(V,\\) each of the points \\(t_\\ell ,\\) \\(\\ell = 1,\\) \\(2,\\) \\(\\cdots,\\) \\(n\\) coincides with one of the points \\(t_j ' : t_{k_{\\ell}} ' = t_{\\ell} ;\\) We have, therefore, the ordering\n\\[0 = k_0 < k_1 < \\cdots < k_n = m ,\\,\\, m \\ge n.\\]\nTherefore, the interval \\([t_\\ell ,\\, t_{\\ell +1}]\\) of \\(V\\) is partitioned into \\(k_{\\ell +1} - k_{\\ell}\\) intervals \\([t_j ' ,\\, t_{j+1} ']\\) of partition \\(W,\\) where \\(j = k_{\\ell} ,\\) \\(k_\\ell +1 ,\\) \\(\\cdots ,\\) \\(k_{\\ell +1} -1 .\\) According to (5), we have for every \\(j\\)\n\\[\\lVert x(t_j ' ) - x(t_i ) \\rVert \\le \\epsilon .\\]\nTherefore\n\\[\\begin{align}\nS(V,\\,x(t)) &= \\sum_{\\ell=0}^{n-1} x(t_\\ell )(t_{\\ell +1} - t_\\ell ) \\\\[6pt]\n&= \\sum_{\\ell =0}^{n-1} x(t_\\ell ) \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} (t_{j+1} ' - t_j ' ),\\\\[8pt]\nS(W,\\,x(t)) &= \\sum_{j=0}^{m-1} x(t_j ' )(t_{j+1} ' - t_j ' ) \\\\[6pt]\n&= \\sum_{\\ell =0}^{n-1} \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} x(t_j ' ) (t_{j+1} ' - t_j ' ),\\\\[8pt]\n\\lVert S(V,\\,x(t)) -  S(W,\\,x(t)) \\rVert \n&=\\left\\lVert \\sum_{\\ell =0}^{n-1} \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} [x(t_\\ell ) - x(t_j ' )] (t_{j+1} ' - t_j ' ) \\right\\rVert \\\\[6pt]\n&\\le \\sum_{\\ell =0}^{n-1} \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} \\lVert x(t_\\ell ) - x(t_j ' ) \\rVert (t_{j+1} ' - t_j ' ) \\\\[6pt]\n&\\le \\sum_{\\ell =0}^{n-1} \\sum_{j=k_\\ell}^{k_{\\ell +1} -1} \\epsilon (t_{j+1} ' - t_j ' ) \\\\[6pt]\n&= \\epsilon (b-a) .\\tag*{\\(\\blacksquare\\)}\n\\end{align}\\]\n<\/p>\n\n\n<\/div>\n\n\n\n\n\n\n<div class=\"box\">\n\n\n<p><span class=\"lemma\">Lemma 3.<\/span>\nLet \\(V_\\tau\\) and \\(V_{\\tau '}\\) be arbitrary \\(\\tau ,\\) \\(\\tau ' \\) partitions, respectively, of the closed interval \\([a,\\,b].\\) Then\n\\[\\lVert S(V_\\tau ,\\, x(t)) - S(V_{\\tau '} ,\\, x(t)) \\rVert \\le (\\epsilon_\\tau + \\epsilon_{\\tau '})(b-a) .\\tag{6}\\]\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof-eng\">\n\n\n<p class=\"proofname\">Proof<\/p>\n\n\n\n\n<p>We can always select a partition \\(W\\) of the interval \\([a,\\,b]\\) which is a refinement of both \\(V_\\tau\\) and \\(V_{\\tau '}.\\) Then, according to (4),\n\\[\\lVert S(V_\\tau ,\\, x(t)) - S(W,\\, x(t)) \\rVert \\le \\epsilon_\\tau (b-a),\\\\[6pt]\n\\lVert S(V_{\\tau '} ,\\, x(t)) - S(W,\\, x(t)) \\rVert \\le \\epsilon_{\\tau '} (b-a),\\]\nfrom which\n\\[\\lVert S(V_\\tau ,\\, x(t)) - S(V_{\\tau '} ,\\, x(t)) \\rVert \\le (\\epsilon_\\tau + \\epsilon_{\\tau '} ) (b-a).\\tag*{\\(\\blacksquare\\)}\\]\n<\/p>\n\n<\/div>\n\n\n\n\n\n<p>Now we prove the theorem 1. For a given \\(\\epsilon ,\\) we consider a sequence of \\(\\tau_n\\)-partitions \\(V_{\\tau_n}\\) of the interval \\([a,\\,b],\\) where \\(\\tau_n \\to 0\\) as \\(n\\to\\infty .\\) Because of Lemma 3, the inequality\n\\[\\lVert S(V_{\\tau_n} ,\\, x(t)) - S(V_{\\tau_{n+1}} ,\\, x(t)) \\lVert \\le (\\epsilon_{\\tau_n} + \\epsilon_{\\tau_{n+1}} )(b-a)\\]\nholds for the corresponding integral sums \\(S(V_{\\tau_n} ,\\, x(t)).\\)<\/p>\n\n\n\n\n\n<p>Since it holds for every \\(\\epsilon,\\) the sequence \\(S(V_{\\tau_n},\\, x(t))\\) is a fundamental sequence and because \\(S(V_{\\tau_n} ,\\, x(t)) \\in E\\) and \\(E\\) is complete, this sequence has a limit \\(S\\) in \\(E.\\)<\/p>\n\n\n\n\n\n<p>Now let \\(V_{\\tau_n}\\) be another sequence of partitions of the interval \\([a,\\,b]\\) where \\(\\tau ' \\to 0.\\) Using the preceding argument as a basis, \\(S(V_{\\tau_n '},\\,x(t))\\) converges to a limit \\(S_1\\) as \\(n\\to\\infty.\\) We shall prove that \\(S = S_1 .\\)<\/p>\n\n\n\n\n\n<p>To do this, we consider the sequence \\(V_{\\tau_1} ,\\) \\(V_{\\tau_1 '},\\) \\(V_{\\tau_2} ,\\) \\(V_{\\tau_2 '} ,\\) \\(\\cdots .\\) The integral sums \\(S(V_{\\tau_n} ,\\, x(t)),\\) \\(S(V_{\\tau_n '} ,\\, x(t)) ,\\) respectively, corresponding to these sequences, form a convergent sequences. If their limits are equal to \\(S_2 ,\\) then they must be equal to the limits \\(S\\) and \\(S_1 .\\) Consequently, \\(S = S_1 = S_2 ,\\) and the theorem is proved.<\/p>\n\n\n\n\n\n<p>We call \\(S\\) the <span class=\"defined\">integral<\/span> of \\(x(t)\\) over \\([a,\\,b]\\) and designate it by\n\\[\\int_a^b x(t) dt.\\]\n<\/p>\n\n\n\n\n\n\n<h3>Properties of the Integral<\/h3>\n\n\n\n\n\n<p>Based on the definition of the integral, the following properties are evident.<\/p>\n\n\n\n\n<p>[1] Integration is additive:\n\\[\\int_a^b [x(t) + y(t)] dt = \\int_a^b x(t) dt + \\int_a^b y(t)dt.\\]\n[2] If \\(\\lambda\\) is a fixed scalar then\n\\[\\int_a^b \\lambda x(t) dt = \\lambda \\int_a^b x(t) dt.\\]\n[3] We also have\n\\[\\left\\lVert \\int_a^b x(t) dt \\right\\rVert \\le \\int_a^b \\lVert x(t) \\rVert dt .\\tag{7}\\]\nFor,\n\\[\\begin{align}\n\\lVert S(V_\\tau ,\\, x(t)) \\rVert \n&= \\left\\lVert \\sum_{i=0}^{n-1} x(t_i )(t_{i+1} - t_i ) \\right\\rVert \\\\[6pt]\n&\\le \\sum_{i=0}^{n-1} \\lVert x(t) \\rVert (t_{i+1} - t_i ) \\\\[6pt]\n&= S(V_\\tau ,\\, \\lVert x(t) \\rVert ). \\tag{8}\n\\end{align}\\]\nAs \\(\\tau \\to 0,\\) the integral sums tend to the integrals\n\\[\\int_a^b x(t) dt \\,\\,\\,\\text{and}\\,\\,\\, \\int_a^b \\lVert x(t) \\rVert dt,\\]\nand the inequality (8) goes over to (7).<\/p>\n\n\n\n\n<p>[4] If there is defined a right-sided(or left-sided) multiplication for the elements \\(x\\in E\\) with elements \\(y\\in E_1 ,\\) then\n\\[\\int_a^b x(t) y dt = \\int_a^b x(t) dt \\,y.\\tag{9}\\]\nFor,\n\\[\\begin{align}\nS(V_\\tau ,\\, x(t)y )\n&= \\sum_{i=0}^{n-1} x(t_i ) y(t_{i+1} - t_i ) \\\\[6pt]\n&=\\left(\\sum_{i=0}^{n-1} x(t_i )(t_{i+1} - t_i ) \\right) y \\\\[6pt]\n&= S(V_\\tau ,\\, x(t)) y \\tag{10}\n\\end{align}\\]\nholds for a \\(\\tau\\)-partition \\(V_\\tau (t_0 ,\\, t_1 ,\\, \\cdots ,\\, t_n ).\\) As \\(\\tau \\to 0,\\) equation (10) goes over to (9).<\/p>\n\n\n\n\n\n<p>Analogously, for left-sided multiplication,\n\\[\\int_a^b yx(t)dt = y\\int_a^b x(t)dt \\tag{9\\( ' \\)}\\]\nholds.<\/p>\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">Example 1.<\/span>\nIf \\(A\\) is a linear operator mapping \\(E\\) into \\(E_1 ,\\) and \\(x(t)\\in E ,\\) then\n\\[\\int_a^b Ax(t)dt = A \\int_a^b x(t)dt.\\]\n<\/p>\n\n<\/div>\n\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">Example 2.<\/span>\nIf \\(A = A(t)\\) is an operator from \\((E \\to E_1 ),\\) which depends continuously on \\(t,\\) and if \\(x\\in E,\\) then\n\\[\\int_a^b A(t) x dt = \\left( \\int_a^b A(t) dt \\right) x.\\]\n<\/p>\n\n<\/div>\n\n\n\n\n\n<p>In the first example, the operator \\(A\\) is a constant left-sided factor and in the second example, \\(x\\) is a constant right-sided factor.<\/p>\n\n\n\n\n\n<p>If, in particular, \\(f\\) is a linear functional from \\(\\overline{E} ,\\) then\n\\[f\\left( \\int_a^b x(t) dt \\right) = \\int_a^b f(x(t)) dt,\\\\[6pt]\n\\int_a^b f(t) (x) dt = \\left( \\int_a^b f(t) dt \\right) (x).\\tag{10\\( ' \\)}\\]\n<\/p>\n\n\n\n\n\n<p>[5] If the function \\(x=x(t),\\) \\(x\\in E,\\) \\(t\\in [a,\\,b]\\) has a continuous derivative with respect to \\(t,\\) \\(x ' (t) = \\frac{d}{dt} x(t) ,\\) then\n\\[\\int_a^b x ' (t) dt = x(b) - x(a) .\\tag{11}\\]\nFor, according to (10'), we obtain, for an arbitrary linear functional \\(L,\\)\n\\[L \\left( \\int_a^b x ' (t) dt \\right) = \\int_a^b L [x ' (t)] dt = \\int_a^b \\frac{d}{dt} \\left\\{ L[x(t)] \\right\\} dt.\\]\n\\([x(t)]\\) is, however, a function of \\(t\\) whose range consists of numbers. Therefore:\n\\[\\int_a^b \\left\\{ L [x(t)] \\right\\} ' dt = L[x(b)] - L[x(a)].\\]\nHence,\n\\[L \\left( \\int_a^b x ' (t) dt \\right) = L [x(b) - x(a)].\\tag{12}\\]\nThe equation (12) holds for an arbitrary linear functional \\(L \\in \\overline{E}.\\) With this, (11) is proved.<\/p>\n\n\n\n\n\n\n\n<h3>Applications to Differential Equations<\/h3>\n\n\n\n\n\n<p>We consider the differential equation\n\\[\\frac{dx}{dt} = f(t,\\,x).\\tag{13}\\]\n\\(x = x(t)\\) and \\(f(t,\\,x)\\) are elements form a normed space \\(E;\\) let \\(t\\in [a,\\,b].\\) We assume that \\(f(t,\\,x)\\) is continuous in \\(t\\) and that, with respect to \\(x,\\) the Lipschitz condition\n\\[\\lVert f(t,\\,x_1 ) - f(t,\\,x_2 ) \\lVert \\le M \\lVert x_1 - x_2 \\rVert \\tag{14}\\]\nis satisfied.<\/p>\n\n\n\n\n\n<p>We designate by \\(C^E [a,\\,b]\\) the space of continuous functions \\(x(t)\\) with \\(t\\in [a,\\,b]\\) and \\(x(t)\\in E.\\) We introduce a norm in \\(C^E [a,\\,b]\\) by\n\\[\\lVert x \\rVert_C = \\max_{a\\le t\\le b} \\lVert x(t) \\rVert .\\]\nBesides \\(E,\\) \\(C^E [a,\\,b]\\) is a complete space. We prove this assertion in the same way as that for the special case \\(E=\\mathbb{R}\\) and \\(C^E [a,\\,b] = C[0,\\,1].\\)<\/p>\n\n\n\n\n\n<p>Besides the equality (13), we consider\n\\[x(t) = x_0 + \\int_{t_0}^t f(t,\\,x(t)) dt,\\,\\, a\\le t_0 \\le t \\le t_0 + \\delta \\le b. \\tag{15}\\]\nWe designate the right side of this equality by \\(A(x).\\) \\(A\\) is an operator with transforms \\(x=x(t)\\)\\(\\in C^E [t_0 ,\\, t_0 + \\delta ]\\) into an element of the same space. We have\n\\[\\begin{align}\n\\lVert A(x) - A(y) \\rVert_C\n&= \\left\\lVert \\int_{t_0}^t [f(t,\\,x(t)) - f(t,\\,y(t))]dt \\right\\rVert_C \\\\[6pt]\n&\\le \\int_{t_0}^{t_0 + \\delta} \\lVert f(t,\\,x(t)) - f(t,\\,y(t)) \\rVert dt.\n\\end{align}\\]\nIt follows, according to (14), that\n\\[\\begin{align}\n\\lVert A(x) - A(y) \\rVert_C &\\le M\\int_{t_0}^{t_0 + \\delta} \\lVert x(t) - y(t) \\rVert dt\\\\[6pt]\n&\\le M \\delta \\max_{t_0 \\le t \\le t_0 + \\delta} \\lVert x(t) - y(t) \\rVert \\\\[6pt]\n&=M \\delta \\lVert x(t) - y(t) \\rVert_C . \\tag{16}\n\\end{align}\\]\nIf \\(M\\delta < 1,\\) then the operator \\(A\\) determines, according to (16), a contracting mapping of the space \\(C^E [a,\\,b]\\) into itself and consequently there exists exactly one solution of (15).<\/p>\n\n\n\n\n\n<p>The equation (15) is equivalent to (13) for the initial value \\(x(t_0 ) - x_0 .\\) Therefore (13) has exactly one solution in the interval \\([t_0 ,\\, t_0 + \\delta ].\\)<\/p>\n\n\n\n\n\n<p>In particular, this equation has for an arbitrary initial value \\(x(a) = x_0\\) exactly one solution \\(x(t)\\) on the interval \\([a,\\,a+\\delta ].\\) \\(x(t)\\) can be continued to the whole interval \\([a,\\,b].\\) For, if \\(a+\\delta < b\\) and \\(x(a+\\delta ) = x_1 ,\\) then we construct by repetition of this process a solution on the interval \\([a+\\delta ,\\, a+2 \\delta ]\\) with the initial value \\(x_1 ,\\) etc.<\/p>\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">Example 3.<\/span>\nIf \\(E\\) is the \\(n\\)-dimensional space, then we obtain the well-known existence theorem for a system of \\(n\\) differential equations.\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">Example 4.<\/span>\nIf \\(E\\) is one of the spaces \\(l_p ,\\) \\(c,\\) \\(m,\\) etc., then we obtain the existence theorem for a solution of the corresponding classes of infinite systems of differential equations.\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">Example 5.<\/span>\nWe consider the integro-differential equation\n\\[\\frac{dy}{dt} = f(t,\\,y) + \\int_a^b K(t,\\,s) y(s) ds. \\tag{17}\\]\nLet \\(f(t,\\,y)\\) satisfy the Lipschitz condition (14) and the kernel \\(K(t,\\,s)\\) be bounded, i.e., \\(\\lvert K(t,\\,s) \\rvert < M_1 .\\)<\/p>\n\n\n\n\n<p>If the right side (17) is designated by \\(F(t,\\,y)\\) we obtain the inequality\n\\[\\begin{align}\n\\lvert F(t,\\,y) - F(t,\\,z) \\rvert\n&\\le M \\lvert y-z \\rvert + M_1 (b-a) \\lvert y-z \\rvert \\\\[6pt]\n&= (M+M_1 (b-a)) \\lvert y-z \\rvert .\n\\end{align}\\]\n\\(F(t,\\,y)\\) satisfies the Lipschitz condition. Consequently, (17) has a unique solution for an arbitrary initial value \\(y=y_0 .\\)\n<\/p>\n\n<\/div>\n\n\n\n\n\n\n\n\n<h3>Reference<\/h3>\n\n\n\n\n<ul class=\"bracket\">\n\n\n<li>L. A. Liusternik ad V. J. Sobolev, \u300eElements of Functional Analysis\u300f (Translated by Anthony E. Labarre), Frederick Ungar Publishing Company(New York), 173-178.\n<\/li>\n\n\n<\/ul>\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc5d0\uc11c\ub294 \ucd94\uc0c1\uacf5\uac04\uc5d0\uc11c\uc758 \uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf4\uc790. \\(E\\)\ub97c \ub178\ub984\ubca1\ud130\uacf5\uac04\uc774\ub77c\uace0 \ud558\uace0, \\(K\\)\ub97c \ub2eb\ud78c\uad6c\uac04 \\([0,\\,1]\\)\uc774\ub77c\uace0 \ud558\uc790. \\(K\\)\uc5d0\uc11c \\(E\\)\ub85c\uc758 \ud568\uc218 \\(x = x(t)\\)\ub97c \uc0dd\uac01\ud558\uc790. (\uc120\ud615\uc778 \uacbd\uc6b0\ubfd0\ub9cc \uc544\ub2c8\ub77c \uc77c\ubc18\uc801\uc778 \ud568\uc218\ub97c \uc0dd\uac01\ud558\uc790.) \uc55e\uc73c\ub85c \uc774\ub7ec\ud55c \ud568\uc218\ub97c \uad6c\uac04 \\([0,\\,1]\\) \uc704\uc5d0\uc11c \uc815\uc758\ub41c \ucd94\uc0c1\ud654\ub41c \ud568\uc218\ub77c\uace0 \ubd80\ub97c \uac83\uc774\ub2e4. \uc774\ub7ec\ud55c \ud568\uc218\uc5d0 \ub300\ud558\uc5ec, \ud574\uc11d\ud559\uc758 \uae30\ubcf8\uc801\uc778 \uc5f0\uc0b0\uc744 \uc815\uc758\ud558\uace0 \uadf8 \uc131\uc9c8\uc744 \uc720\ub3c4\ud558\uc790. \uc801\ubd84\uc758 \uc815\uc758 \ub2eb\ud78c\uad6c\uac04 \\([a,\\,b]\\)\ub97c \ub2eb\ud78c \ubd80\ubd84\uad6c\uac04 \\([t_i ,\\, t_{i+1}]\\)\ub85c \ub098\ub208 \ubd84\ud560 \\(A = A[t_0 ,\\, t_1 ,\\, \\cdots&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"categories":[70],"tags":[82,93,95,96,94,90,97,98,91,92],"class_list":["post-2584","post","type-post","status-publish","format-standard","hentry","category-functional-analysis","tag-abstract-function","tag-differential-equation","tag-functional-analysis","tag-integral","tag-integral-sum","tag-integration","tag-integro-differential-equation","tag-lipschitz-condition","tag-partition","tag-refinement"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2584","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"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=2584"}],"version-history":[{"count":38,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2584\/revisions"}],"predecessor-version":[{"id":9466,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2584\/revisions\/9466"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2584"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2584"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2584"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}