{"id":2415,"date":"2019-03-05T12:21:05","date_gmt":"2019-03-05T03:21:05","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=2415"},"modified":"2019-07-25T22:36:20","modified_gmt":"2019-07-25T13:36:20","slug":"stone-weierstrass-theorem","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/stone-weierstrass-theorem\/","title":{"rendered":"\uc2a4\ud1a4-\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \uc815\ub9ac"},"content":{"rendered":"

\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4\uc758 \uadfc\uc0ac \uc815\ub9ac\uc5d0 \uc758\ud558\uba74 \\(f\\)\uac00 \ub2eb\ud78c \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc5f0\uc18d\ud568\uc218\uc77c \ub54c \\([a,\\,b]\\) \uc704\uc5d0\uc11c \\(f\\)\uc5d0 \uade0\ub4f1\uc218\ub834\ud558\ub294 \ub2e4\ud56d\ud568\uc218\uc5f4\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc5ec\uae30\uc11c\ub294 \ub354 \uc77c\ubc18\uc801\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

\n

\uc815\uc758 1. (\ub300\uc218; algebra)<\/span><\/p>\n

\\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(C(X)\\)\uac00 \uc815\uc758\uc5ed\uc774 \\(X\\)\uc778 \uc5f0\uc18d \uc2e4\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc774\ub77c\uace0 \ud558\uc790. \\(\\mathcal{A}\\)\uac00 \\(C(X)\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \ub450 \uc870\uac74<\/p>\n

    \n
  1. \\(f,\\,g\\in\\mathcal{A}\\)\uc77c \ub54c \\(f+g \\in \\mathcal{A},\\) \\(fg\\in\\mathcal{A}\\)<\/li>\n
  2. \\(f\\in\\mathcal{A},\\) \\(c\\in\\mathbb{R}\\)\uc77c \ub54c \\(cf\\in\\mathcal{A}\\)<\/li>\n<\/ol>\n

    \ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(\\mathcal{A}\\)\ub97c \\(C(X)\\)\uc758 \uc2e4\ud568\uc218 \ub300\uc218<\/span>(real function algebra) \ub610\ub294 \uac04\ub2e8\ud788 \ub300\uc218<\/span>(algebra)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

    \\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(f\\in C(X)\\)\uc77c \ub54c \\(f\\)\uc758 \uade0\ub4f1\ub178\ub984<\/span>(uniform norm)\uc744
    \n\\[\\lVert f \\rVert := \\sup_{x\\in X} \\lvert f(x) \\rvert\\]
    \n\ub85c \uc815\uc758\ud55c\ub2e4. \uadf8\ub9ac\uace0 \ub450 \ud568\uc218 \\(f,\\,g\\in C(X)\\)\uc758 \uac70\ub9ac<\/span>\ub97c
    \n\\[\\operatorname{dist}(f,\\,g) := \\lVert f-g \\rVert \\]
    \n\ub85c \uc815\uc758\ud55c\ub2e4. \ub9cc\uc57d \\(X\\)\uac00 \ucef4\ud329\ud2b8 \uacf5\uac04\uc774\uba74 \\(\\operatorname{dist}\\)\ub294 \\(C(X)\\)\uc758 \uac70\ub9ac\ud568\uc218\uac00 \ub41c\ub2e4.<\/p>\n

    \\(\\left\\{ f_n \\right\\}\\)\uc774 \\(C(X)\\)\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc218\uc5f4\uc77c \ub54c \\(\\left\\{f_n \\right\\}\\)\uc774 \\(f\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\ub294 \uac83\uc740 \\(\\lVert f_n – f \\rVert \\,\\to\\, 0\\)\uc778 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4. \ub530\ub77c\uc11c \\(C(X)\\)\uc5d0\uc11c\uc758 \uc218\ub834\uc740 \ud568\uc218\uc5f4\uc758 \uade0\ub4f1\uc218\ub834\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n

    \n

    \uc815\uc758 2. (\ub300\uc218\uc758 \ub2eb\ud798\uacfc \uade0\ub4f1\uc870\ubc00\uc131)<\/span><\/p>\n

    \\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(\\mathcal{A}\\subseteq C(X)\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

      \n
    1. \ub9cc\uc57d \\(\\mathcal{A}\\)\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ud568\uc218\uc5f4 \\(\\left\\{f_n \\right\\}\\)\uc774 \uc218\ub834\ud560 \ub54c\ub9c8\ub2e4 \uadf8 \uadf9\ud55c\ud568\uc218\uac00 \\(\\mathcal{A}\\)\uc5d0 \uc18d\ud558\uba74 \\(\\mathcal{A}\\)\ub294 \ub2eb\ud600\uc788\ub2e4<\/span>(closed)\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n
    2. \uc784\uc758\uc758 \uc591\uc218 \\(\\epsilon\\)\uacfc \\(f\\in C(X)\\)\uc5d0 \ub300\ud558\uc5ec \\(g\\in\\mathcal{A}\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\lVert f-g \\rVert < \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(\\mathcal{A}\\)\ub294 \\(C(X)\\)\uc5d0\uc11c \uade0\ub4f1\uc870\ubc00\ud558\ub2e4<\/span>(uniformly dense)\uace0 \ub9d0\ud55c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

      \uade0\ub4f1\uc218\ub834\ud558\ub294 \uc5f0\uc18d\ud568\uc218\uc5f4\uc758 \uadf9\ud55c\ud568\uc218\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \uba85\ubc31\ud788 \\(C(X)\\)\ub294 \ub2eb\ud78c \uc9d1\ud569\uc774\ub2e4.<\/p>\n

      \n

      \ubcf4\uc870\uc815\ub9ac 1.<\/span>
      \n\\(X\\)\uac00 \ucef4\ud329\ud2b8 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(\\mathcal{A}\\)\uac00 \\(C(X)\\)\uc758 \ub2eb\ud78c \ub300\uc218\uc774\uba70 \uc0c1\uc218\ud568\uc218\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\(C(X)\\)\uc758 \uc6d0\uc18c \\(f,\\) \\(g\\)\uc5d0 \ub300\ud558\uc5ec \\[f \\wedge g := \\min \\left\\{ f,\\,g \\right\\} ,\\\\[6pt] f\\vee g := \\max \\left\\{ f,\\,g \\right\\}\\]\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\wedge\\)\uc640 \\(\\vee\\)\ub294 \\(\\mathcal{A}\\)\uc758 \uc774\ud56d\uc5f0\uc0b0\uc774\ub2e4. \uc989 \\(f,\\,g\\in \\mathcal{A}\\)\uc774\uba74 \\(f\\wedge g \\in \\mathcal{A},\\) \\(f\\vee g \\in \\mathcal{A}\\)\uc774\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \uc784\uc758\uc758 \\(f,\\,g\\in C(X)\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[f\\wedge g = \\frac{f+g-|f-g|}{2} ,\\\\[6pt] f\\vee g = \\frac{f+g+|f-g|}{2}\\]
      \n\uc774\uba70 \\(\\mathcal{A}\\)\uac00 \ub300\uc218\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \\(f\\in\\mathcal{A}\\)\uc5d0 \ub300\ud558\uc5ec \\(|f|\\in\\mathcal{A}\\)\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uba74 \ucda9\ubd84\ud558\ub2e4.<\/p>\n

      \\(f\\in\\mathcal{A}\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(\\lVert f \\rVert =0\\)\uc774\uba74 \\(|f| =0 \\in\\mathcal{A}\\)\ub85c\uc11c \uc99d\uba85\uc774 \ub05d\ub09c\ub2e4.<\/p>\n

      \uc774\uc81c \\(\\lVert f \\rVert = M \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
      \n\\[\\begin{align}
      \n|t| &= (1-(1-t^2 ))^{1\/2} \\\\[6pt]
      \n&= 1 – \\frac{1}{2}(1-t^2 )-\\frac{1}{2\\cdot 4} (1-t^2 )^2 \\\\[6pt]&\\quad – \\sum_{k=3}^{\\infty} \\frac{1\\cdot 3\\cdot 5 \\cdots (2k-3)}{2 \\cdot 4 \\cdot 6 \\cdots (2k)} (1-t^2 )^k
      \n\\end{align}\\]
      \n\uc740 \uc5f4\ub9b0\uad6c\uac04 \\((-\\sqrt{2} ,\\, \\sqrt{2} )\\)\uc774 \ucef4\ud329\ud2b8 \ubd80\ubd84\uc9d1\ud569 \uc704\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4. \ub530\ub77c\uc11c \uc704 \uc774\ud56d\uae09\uc218\ub294 \\([-1,\\,1]\\)\uc5d0\uc11c \uade0\ub4f1\uc218\ub834\ud55c\ub2e4.
      \n\\[\\begin{align}
      \nP_n (t) &:= 1 – \\frac{1}{2} (1-t^2 ) – \\frac{1}{2\\cdot 4} (1-t^2 )^2 \\\\[6pt]
      \n&\\quad – \\sum_{k=3}^{n} \\frac{1\\cdot 3\\cdot 5 \\cdots (2k-3)}{2\\cdot 4 \\cdot 6 \\cdots (2k)} (1-t^2 )^k
      \n\\end{align}\\]
      \n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc790\uc5f0\uc218 \\(N\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n \\ge N ,\\) \\(t\\in [-1,\\,1]\\)\uc77c \ub54c\ub9c8\ub2e4 \\(\\left\\lvert P_n (t) – |t| \\right\\rvert < \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(x\\in X\\)\uac00 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0\n\\[g_n (x) := P_n \\left( \\frac{f(x)}{M} \\right)\\]\n\ub77c\uace0 \ud558\uc790. \\(\\mathcal{A}\\)\uac00 \ub300\uc218\uc774\uace0 \uc0c1\uc218\ud568\uc218\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c0\ubbc0\ub85c \\(g_n \\in\\mathcal{A}\\)\uc774\ub2e4. \\(t := f(x)\/M\\)\uc774\ub77c\uace0 \ud558\uba74 \\(M = \\lVert f \\rVert > 0\\)\uc774\ubbc0\ub85c \\(t\\in [-1,\\,1]\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(n\\ge N ,\\) \\(x\\in X\\)\uc77c \ub54c\ub9c8\ub2e4
      \n\\[\\left\\lvert g_n (x) – \\left\\lvert \\frac{f(x)}{M} \\right\\rvert \\right\\rvert
      \n= \\left\\lvert P_n (t) – |t| \\right\\rvert < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(Mg_n \\in\\mathcal{A}\\)\uc774\uace0, \\(n\\,\\to\\,\\infty\\)\uc77c \ub54c \\(Mg_n \\,\\rightrightarrows \\, |f|\\)\uc774\ub2e4.<\/p>\n

      \uadf8\ub7f0\ub370 \\(\\mathcal{A}\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc774\ubbc0\ub85c \\(|f|\\in\\mathcal{A}\\)\uc774\ub2e4.
      \n<\/span><\/p>\n<\/div>\n

      \n

      \uc815\uc758 3. (\uc810\uc758 \ubd84\ub9ac)<\/span><\/p>\n

      \\(\\mathcal{A}\\)\uac00 \\(C(X)\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(x\\ne y\\)\uc778 \uc784\uc758\uc758 \\(x,\\,y\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\in\\mathcal{A}\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(f(x) \\ne f(y)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(\\mathcal{A}\\)\ub294 \\(X\\)\uc758 \uc810\uc744 \ubd84\ub9ac\uc2dc\ud0a8\ub2e4<\/span>(separate points of X)\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n<\/div>\n

      \uc774\uc81c \ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4\uc758 \uadfc\uc0ac \uc815\ub9ac\ub97c \uc77c\ubc18\ud654\ud55c \ub2e4\uc74c \uc815\ub9ac\ub97c \uc99d\uba85\ud558\uc790.<\/p>\n

      \n

      \uc815\ub9ac 1. (\uc2a4\ud1a4-\ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \uc815\ub9ac; Stone-Weierstrass theorem)<\/span><\/p>\n

      \\(X\\)\uac00 \ucef4\ud329\ud2b8 \uac70\ub9ac\uacf5\uac04\uc774\ub77c\uace0 \ud558\uc790. \\(\\mathcal{A}\\)\uac00 \\(C(X)\\)\uc758 \ub300\uc218\uc774\uace0 \\(X\\)\uc758 \uc810\uc744 \ubd84\ub9ac\uc2dc\ud0a4\uba70 \uc0c1\uc218\ud568\uc218\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c0\uba74 \\(\\mathcal{A}\\)\ub294 \\(C(X)\\)\uc5d0\uc11c \uade0\ub4f1\uc870\ubc00\ud558\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\(X\\)\uc758 \uc11c\ub85c \ub2e4\ub978 \uc810 \\(x,\\) \\(y\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(a,\\) \\(b\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \\(\\mathcal{A}\\)\uac00 \\(X\\)\uc758 \uc810\uc744 \ubd84\ub9ac\uc2dc\ud0a4\ubbc0\ub85c \\(g(x) \\ne g(y)\\)\uc778 \ud568\uc218 \\(g\\in\\mathcal{A}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(\\mathcal{A}\\)\uac00 \uc0c1\uc218\ud568\uc218\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c0\ubbc0\ub85c \ud568\uc22b\uac12\uc774 \\(g(x)\\)\uc640 \uac19\uc740 \uc0c1\uc218\ud568\uc218\uc640 \ud568\uc22b\uac12\uc774 \\(g(y)\\)\uc640 \uac19\uc740 \uc0c1\uc218\ud568\uc218\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c4\ub2e4. \ub354\uc6b1\uc774 \\(\\mathcal{A}\\)\uac00 \ub300\uc218\uc774\ubbc0\ub85c
      \n\\[f(t) := a \\frac{g(t) – g(y)}{g(x) – g(y)} + b \\frac{g(t) – g(x)}{g(y) – g(x)}\\]
      \n\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\in\\mathcal{A}\\)\uc774\ub2e4. \ub610\ud55c \\(f(x) = a,\\) \\(f(y) = b\\)\uc774\ubbc0\ub85c \ub2e4\uc74c\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4:<\/p>\n

      \n

      \ubcf4\uc870\uc815\ub9ac 2.<\/span><\/p>\n

      \\(x,\\,y\\in X\\)\uc774\uace0 \\(x \\ne y\\)\uc774\uba70 \\(a,\\,b\\in\\mathbb{R}\\)\uc774\uba74 \\(f(x) =a,\\) \\(f(y)=b\\)\uc778 \\(f\\in\\mathcal{A}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n<\/div>\n

      \\(\\mathcal{B}\\)\ub97c \\(\\mathcal{A}\\)\uc758 \ud3d0\ud3ec(closure), \uc989 \\(\\mathcal{A}\\)\uc758 \uc810\ub4e4\ub85c \uad6c\uc131\ub41c \ud568\uc218\uc5f4\uc758 \uadf9\ud55c\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc774\ub77c\uace0 \ud558\uc790. \\(\\mathcal{A}\\)\uac00 \ub300\uc218\uc774\ubbc0\ub85c \\(\\mathcal{B}\\)\ub3c4 \ub300\uc218\uc774\ub2e4. \\(\\mathcal{B}\\)\ub294 \ub2eb\ud78c\uc9d1\ud569\uc774\ubbc0\ub85c \ubcf4\uc870\uc815\ub9ac 1\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(f,\\) \\(g\\in\\mathcal{B}\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\wedge g\\in \\mathcal{B}\\)\uc774\uace0 \\(f\\vee g\\in \\mathcal{B}\\)\uc774\ub2e4.<\/p>\n

      \ud568\uc218 \\(F\\in C(X)\\)\uc640 \uc591\uc218 \\(\\epsilon\\)\uc774 \uc784\uc758\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(G\\in\\mathcal{B}\\)\uac00 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[F(x) – \\epsilon < G(x) < F(x) + \\epsilon \\tag{1}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.<\/p>\n

      \\(x_0 \\in X\\)\ub77c\uace0 \ud558\uc790. \\(y\\ne x_0\\)\uc778 \uac01 \uc810 \\(y\\)\uc5d0 \ub300\ud558\uc5ec, \ubcf4\uc870\uc815\ub9ac 2\uc5d0 \uc758\ud558\uc5ec \\(f_y \\in \\mathcal{A}\\)\uac00 \uc874\uc7ac\ud558\uc5ec
      \n\\[f_y (x_0 ) = F(x_0 ) ,\\, f_y (y) = F(y)\\]
      \n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(f_y\\)\uc640 \\(F\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c
      \n\\[V_y := \\left\\{ x\\in X \\,\\vert\\, f_y (x) < F(x) + \\epsilon \\right\\}\\]\n\uc740 \uc5f4\ub9b0 \uc9d1\ud569\uc774\ub2e4. \uc784\uc758\uc758 \\(y\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(x_0 \\in V_y ,\\) \\(y\\in V_y \\)\uc774\ubbc0\ub85c\n\\[X = \\bigcup_{y\\ne x_0} V_y\\]\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \\(X\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569\uc774\ubbc0\ub85c \\(X\\)\uc758 \uc810 \\(y_1 ,\\) \\(y_2 ,\\) \\(\\cdots ,\\) \\(y_N\\)\uc774 \uc874\uc7ac\ud558\uc5ec\n\\[X = V_{y_1} \\cup V_{y_2} \\cup \\cdots \\cup V_{y_N}\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \\(f_j := f_{y_j}\\)\uc774\uace0\n\\[g_{x_0} = f_1 \\wedge f_2 \\wedge \\cdots \\wedge f_N\\]\n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(g_{x_0}\\in \\mathcal{B}\\)\uc774\uba70\n\\[g_{x_0} (x_0 ) = F(x_0 ) \\wedge F(x_0 ) \\wedge \\cdots \\wedge F(X_0 ) = F(x_0 )\\]\n\uc774\ub2e4. \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(N\\) \uc774\ud558\uc758 \uc790\uc5f0\uc218 \\(k\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(x\\in V_{y_k}\\)\uc774\ubbc0\ub85c\n\\[g_{x_0} (x) \\le f_k (x) < F(x) + \\epsilon \\tag{2}\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub85c\uc368 (1)\uc758 \uc624\ub978\ucabd \ubd80\ub4f1\uc2dd\uc774 \uc99d\uba85\ub418\uc5c8\ub2e4.\n\\[W_{x_j} := \\left\\{ x\\in X \\,\\vert\\, g_{x_j} (x) > F(x) – \\epsilon \\right\\}\\]
      \n\uc774\ub77c\uace0 \ud558\uace0 \\(X\\)\ub97c \ub36e\ub294 \uc720\ud55c \uc5f4\ub9b0\ub36e\uac1c
      \n\\[\\left\\{W_{x_j} \\,\\vert\\, j =1 ,\\,2,\\,\\cdots,\\,M \\right\\}\\]
      \n\uc744 \uc774\uc6a9\ud558\uc5ec \uc704 \ub17c\uc758\ub97c \ubc18\ubcf5\ud558\uc790.
      \n\\[g_j := g_{x_j} ,\\quad \\, G := g_1 \\vee g_2 \\vee \\cdots \\vee g_M\\]
      \n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 (2)\uc5d0 \uc758\ud558\uc5ec \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(G(x) < F(x) + \\epsilon\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7f0\ub370 \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec \\(j\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(x\\in W_{x_j}\\)\uc774\ubbc0\ub85c \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec\n\\[G(x) \\ge g_j (x) \\ge F(x) - \\epsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub530\ub77c\uc11c \uc784\uc758\uc758 \\(x\\in X\\)\uc5d0 \ub300\ud558\uc5ec (1)\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/span><\/p>\n<\/div>\n

      \n

      \ucc38\uace0.<\/span>
      \n\ub9cc\uc57d \\(C(X)\\)\uac00 \\(X\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \ubcf5\uc18c\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc774\uace0 \\(\\mathcal{A}\\)\uac00 \\(C(X)\\)\uc758 \ubcf5\uc18c\ud568\uc218 \ub300\uc218\uc774\uba74 \uc815\ub9ac 1\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ub098 \ub9cc\uc57d \uc784\uc758\uc758 \\(f\\in\\mathcal{A}\\)\uc5d0 \ub300\ud558\uc5ec \ucf24\ub808\ubcf5\uc18c\ud568\uc218\uac00 \\(\\mathcal{A}\\)\uc5d0 \uc18d\ud55c\ub2e4\ub294 \uc870\uac74, \uc989 \\(\\overline{f} \\in\\mathcal{A}\\)\ub77c\ub294 \uc870\uac74\uc774 \ucd94\uac00\ub418\uba74 \uc815\ub9ac 1\uacfc \uac19\uc740 \uacb0\ub860\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\(\\mathcal{A}\\)\uc5d0 \uc18d\ud558\ub294 \uc2e4\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc744 \\(\\mathcal{A}_{\\mathbb{R}}\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(x_1 \\ne x_2\\)\uc774\uba74 \\(f(x_1 ) =1,\\) \\(f(x_2 ) =0\\)\uc778 \ud568\uc218 \\(f\\in\\mathcal{A}\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(u\\)\uc640 \\(v\\)\uac00 \uc2e4\ud568\uc218\uc774\uace0 \\(f = u + \\mathbf{i} v\\)\ub77c\uace0 \ud558\uba74
      \n\\[u = \\frac{f+\\overline{f}}{2} \\in\\mathcal{A}\\]
      \n\uc774\ub2e4. \ub530\ub77c\uc11c \\(0 = u(x_2 ) \\ne u(x_1 ) = 1\\)\uc774\ubbc0\ub85c \\(\\mathcal{A}_{\\mathbb{R}}\\)\ub294 \\(X\\)\uc758 \uc810\uc744 \ubd84\ub9ac\uc2dc\ud0a8\ub2e4. \ub610\ud55c \uba85\ubc31\ud788 \\(\\mathcal{A}_{\\mathbb{R}}\\)\ub294 \ubaa8\ub4e0 \uc0c1\uc218\ud568\uc218\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c4\ub2e4.<\/p>\n

      \\(\\mathcal{A}_{\\mathbb{R}}\\)\uac00 \uc815\ub9ac 1\uc758 \uac00\uc815\uc744 \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c \\(\\mathcal{A}_{\\mathbb{R}}\\)\ub294 \\(C(X)\\)\uc5d0\uc11c \uade0\ub4f1\uc870\ubc00\ud558\ub2e4. \ub9cc\uc57d \\(\\phi\\)\uac00 \\(X\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \ubcf5\uc18c\ud568\uc218\uc774\uba74 \\(\\phi\\)\uc758 \uc2e4\uc218\ubd80\uc640 \ud5c8\uc218\ubd80\ub294 \uac01\uac01 \\(C(X)\\)\uc5d0 \uc18d\ud558\ub294 \uc2e4\ud568\uc218\uc774\ubbc0\ub85c \\(\\phi\\)\uc758 \uc2e4\uc218\ubd80\uc5d0 \uc218\ub834\ud558\ub294 \\(\\mathcal{A}_{\\mathbb{R}}\\)\uc758 \ud568\uc218\uc5f4\uacfc \\(\\phi\\)\uc758 \ud5c8\uc218\ubd80\uc5d0 \uc218\ub834\ud558\ub294 \\(\\mathcal{A}_{\\mathbb{R}}\\)\uc758 \ud568\uc218\uc5f4\uc744 \uad6c\uc131\ud560 \uc218 \uc788\ub2e4.<\/p>\n

      \ub530\ub77c\uc11c \\(\\mathcal{A}\\)\ub294 \\(C(X)\\)\uc5d0\uc11c \uade0\ub4f1\uc870\ubc00\ud558\ub2e4.
      \n<\/span><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

      \ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4\uc758 \uadfc\uc0ac \uc815\ub9ac\uc5d0 \uc758\ud558\uba74 \\(f\\)\uac00 \ub2eb\ud78c \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc5f0\uc18d\ud568\uc218\uc77c \ub54c \\([a,\\,b]\\) \uc704\uc5d0\uc11c \\(f\\)\uc5d0 \uade0\ub4f1\uc218\ub834\ud558\ub294 \ub2e4\ud56d\ud568\uc218\uc5f4\uc774 \uc874\uc7ac\ud55c\ub2e4. \uc5ec\uae30\uc11c\ub294 \ub354 \uc77c\ubc18\uc801\uc778 \uacbd\uc6b0\ub97c \uc0b4\ud3b4\ubcf4\uc790. \uc815\uc758 1. (\ub300\uc218; algebra) \\(X\\)\uac00 \uac70\ub9ac\uacf5\uac04\uc774\uace0 \\(C(X)\\)\uac00 \uc815\uc758\uc5ed\uc774 \\(X\\)\uc778 \uc5f0\uc18d \uc2e4\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc774\ub77c\uace0 \ud558\uc790. \\(\\mathcal{A}\\)\uac00 \\(C(X)\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uba70 \ub450 \uc870\uac74 \\(f,\\,g\\in\\mathcal{A}\\)\uc77c \ub54c \\(f+g \\in \\mathcal{A},\\) \\(fg\\in\\mathcal{A}\\) \\(f\\in\\mathcal{A},\\) \\(c\\in\\mathbb{R}\\)\uc77c \ub54c \\(cf\\in\\mathcal{A}\\) \ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(\\mathcal{A}\\)\ub97c \\(C(X)\\)\uc758 \uc2e4\ud568\uc218 \ub300\uc218(real function algebra) \ub610\ub294 \uac04\ub2e8\ud788 \ub300\uc218(algebra)\ub77c\uace0…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[50],"tags":[],"class_list":["post-2415","post","type-post","status-publish","format-standard","hentry","category-mathematical-analysis"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2415","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=2415"}],"version-history":[{"count":17,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2415\/revisions"}],"predecessor-version":[{"id":2432,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/2415\/revisions\/2432"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=2415"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=2415"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=2415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}