. "Operator nazywa si\u0119 operatorem liniowym ograniczonym je\u017Celi: \n* jest operatorem liniowym, \n* i s\u0105 przestrzeniami unormowanymi, \n* istnieje pewna liczba nieujemna taka \u017Ce dla ka\u017Cdego nale\u017C\u0105cego do spe\u0142niony jest warunek Operator ograniczony nie jest w og\u00F3lno\u015Bci funkcj\u0105 ograniczon\u0105; wymaga\u0142oby to by norma by\u0142a mniejsza od pewnej liczby dla wszystkich wektor\u00F3w tj. co zachodzi jedynie, gdy operator jest funkcj\u0105 ograniczon\u0105, np. Operator liniowy ograniczony jest jednak zawsze funkcj\u0105 lokalnie ograniczon\u0105, co oznacza, \u017Ce dla ka\u017Cdego wektora istnieje otoczenie, w kt\u00F3rym warto\u015Bci operatora s\u0105 liczbami sko\u0144czonymi, gdzie nale\u017Cy do otoczenia wektora Norm\u0105 operatora nazywa si\u0119 najmniejsz\u0105 liczb\u0119 spe\u0142niaj\u0105c\u0105 warunek podany w definicji tego operatora."@pl . . . . "In analisi funzionale un operatore limitato \u00E8 un operatore tra due spazi metrici e tale per cui, comunque si scelga un sottoinsieme limitato , l'insieme \u00E8 un sottoinsieme limitato di . Un operatore lineare continuo limitato tra spazi vettoriali normati \u00E8 una funzione tale per cui il rapporto tra la norma dell'immagine di un vettore e la norma del vettore stesso sia limitato dallo stesso numero per ogni vettore non nullo del dominio. In particolare, un operatore lineare \u00E8 limitato se e solo se \u00E8 continuo."@it . . . . . . . . . . . . . "Em matem\u00E1tica e, em especial, em an\u00E1lise funcional um operador linear limitado \u00E9 um operador linear L entre espa\u00E7os normados em que a norma de um vetor x est\u00E1 limitado (em sentido a ser definido precisamente abaixo) pela norma de x."@pt . . "\u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043E\u0433\u0440\u0430\u043D\u0438\u0447\u0435\u043D\u043D\u044B\u043C, \u0435\u0441\u043B\u0438 \u043A\u0430\u0436\u0434\u043E\u0435 \u043E\u0433\u0440\u0430\u043D\u0438\u0447\u0435\u043D\u043D\u043E\u0435 \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u043E \u0438\u0441\u0445\u043E\u0434\u043D\u043E\u0433\u043E \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0447\u0435\u0441\u043A\u043E\u0433\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u043E\u043D \u043F\u0435\u0440\u0435\u0432\u043E\u0434\u0438\u0442 \u0432 \u043E\u0433\u0440\u0430\u043D\u0438\u0447\u0435\u043D\u043D\u043E\u0435 \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u043E \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0447\u0435\u0441\u043A\u043E\u0433\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 . \u041F\u0440\u0438\u0432\u0435\u0434\u0451\u043D\u043D\u043E\u0435 \u0432\u044B\u0448\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u043E\u0442\u043D\u043E\u0441\u0438\u0442\u0441\u044F \u043A\u0430\u043A \u043A \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u043C, \u0442\u0430\u043A \u0438 \u043A ."@ru . . . "Proof"@en . "Beschr\u00E4nkter Operator"@de . . . . . . . . . . . . . . . . "Un operador lineal acotado u operador acotado es una aplicaci\u00F3n lineal definida sobre un espacio vectorial normado tal que la norma de sus valores puede acotarse. M\u00E1s precisamente, la aplicaci\u00F3n lineal es un operador acotado si y solo s\u00ED:"@es . . . . . . . . . . . . . "Operator nazywa si\u0119 operatorem liniowym ograniczonym je\u017Celi: \n* jest operatorem liniowym, \n* i s\u0105 przestrzeniami unormowanymi, \n* istnieje pewna liczba nieujemna taka \u017Ce dla ka\u017Cdego nale\u017C\u0105cego do spe\u0142niony jest warunek Operator ograniczony nie jest w og\u00F3lno\u015Bci funkcj\u0105 ograniczon\u0105; wymaga\u0142oby to by norma by\u0142a mniejsza od pewnej liczby dla wszystkich wektor\u00F3w tj. co zachodzi jedynie, gdy operator jest funkcj\u0105 ograniczon\u0105, np. gdzie nale\u017Cy do otoczenia wektora Norm\u0105 operatora nazywa si\u0119 najmniejsz\u0105 liczb\u0119 spe\u0142niaj\u0105c\u0105 warunek podany w definicji tego operatora."@pl . . . . . . . . . . . . . "\u6709\u754C\u7B97\u5B50"@zh . . . . . . . . . . . . . . . . . . . . . . "Op\u00E9rateur born\u00E9"@fr . . . . . . . . "\u95A2\u6570\u89E3\u6790\u5B66\u306B\u304A\u3044\u3066\u6709\u754C\uFF08\u7DDA\u5F62\uFF09\u4F5C\u7528\u7D20\uFF08\u3086\u3046\u304B\u3044\u3055\u3088\u3046\u305D\u3001\u82F1: Bounded\u3008linear\u3009operator\uFF09\u3068\u306F\u3001\u4E8C\u3064\u306E\u30CE\u30EB\u30E0\u7A7A\u9593 X \u304A\u3088\u3073 Y \u306E\u9593\u306E\u7DDA\u578B\u4F5C\u7528\u7D20 L \u3067\u3042\u3063\u3066\u3001X \u306B\u542B\u307E\u308C\u308B\u30BC\u30ED\u3067\u306A\u3044\u3059\u3079\u3066\u306E\u30D9\u30AF\u30C8\u30EB v \u306B\u5BFE\u3057\u3066 L(v) \u306E\u30CE\u30EB\u30E0\u3068 v \u306E\u30CE\u30EB\u30E0\u306E\u6BD4\u304C\u3001v \u306B\u4F9D\u5B58\u3057\u306A\u30441\u3064\u306E\u6570\u306B\u3088\u3063\u3066\u4E0A\u304B\u3089\u8A55\u4FA1\u3055\u308C\u308B\u3088\u3046\u306A\u3082\u306E\u306E\u3053\u3068\u3092\u8A00\u3046\u3002\u8A00\u3044\u63DB\u3048\u308B\u3068\u3001\u6B21\u3092\u6E80\u305F\u3059\u7DDA\u578B\u4F5C\u7528\u7D20 L \u306E\u3053\u3068\u3092\u3001\u6709\u754C\u4F5C\u7528\u7D20\u3068\u8A00\u3046: \u3053\u3053\u3067 \u306F X \u304C\u5099\u3048\u308B\u30CE\u30EB\u30E0\u3067\u3042\u308B\uFF08 \u3082\u540C\u69D8\uFF09\uFF0E\u4E0A\u8A18\u306E\u6B63\u5B9A\u6570 M \u306E\u4E0B\u9650\u306F L \u306E\u4F5C\u7528\u7D20\u30CE\u30EB\u30E0\u3068\u547C\u3070\u308C\u3001 \u3068\u8A18\u8FF0\u3055\u308C\u308B\u3002 X \u304B\u3089 Y \u3078\u306E\u6709\u754C\u4F5C\u7528\u7D20\u5168\u4F53\u306E\u96C6\u5408\u3092 \u3068\u3057\u3066\uFF0C\u306B\u5BFE\u3057\u3066 \u306B\u3088\u3063\u3066\u4F5C\u7528\u7D20\u30CE\u30EB\u30E0\u3092\u8868\u3059\u3053\u3068\u3082\u3042\u308B\uFF0E \u4E00\u822C\u7684\u306B\u3001\u6709\u754C\u4F5C\u7528\u7D20\u306F\u6709\u754C\u95A2\u6570\u3067\u306F\u306A\u3044\u3002\u5F8C\u8005\u306F\u3001\u3059\u3079\u3066\u306E v \u306B\u5BFE\u3057 L(v) \u306E\u30CE\u30EB\u30E0\u304C\u4E0A\u304B\u3089\u8A55\u4FA1\u3055\u308C\u3066\u3044\u308B\u5FC5\u8981\u304C\u3042\u308B\u304C\u3001\u3053\u308C\u306F L \u304C\u96F6\u4F5C\u7528\u7D20\u3067\u306A\u3044\u3068\u8D77\u3053\u308A\u5F97\u306A\u3044\u3002\u6709\u754C\u4F5C\u7528\u7D20\u306F\u3067\u3042\u308B\u3002 \u7DDA\u5F62\u4F5C\u7528\u7D20\u304C\u6709\u754C\u3067\u3042\u308B\u3053\u3068\u3068\u3001\u9023\u7D9A\u3067\u3042\u308B\u3053\u3068\u306F\u5FC5\u8981\u5341\u5206\u3067\u3042\u308B\u3002"@ja . . "\uD568\uC218\uD574\uC11D\uD559\uC5D0\uC11C \uC720\uACC4 \uC791\uC6A9\uC18C(\u6709\u754C\u4F5C\u7528\u7D20, \uC601\uC5B4: bounded operator)\uB294 \uC720\uACC4 \uC9D1\uD569\uC744 \uD56D\uC0C1 \uC720\uACC4 \uC9D1\uD569\uC5D0 \uB300\uC751\uC2DC\uD0A4\uB294, \uB450 \uC704\uC0C1 \uBCA1\uD130 \uACF5\uAC04 \uC0AC\uC774\uC758 \uC120\uD615 \uBCC0\uD658\uC774\uB2E4. \uB450 \uB178\uB984 \uACF5\uAC04 \uC0AC\uC774\uC758 \uACBD\uC6B0, \uC720\uACC4 \uC791\uC6A9\uC18C\uC758 \uAC1C\uB150\uC740 \uC5F0\uC18D \uC120\uD615 \uBCC0\uD658\uC758 \uAC1C\uB150\uACFC \uC77C\uCE58\uD55C\uB2E4."@ko . . "1120542068"^^ . . . . . . . . "Operatore limitato"@it . . . "Suppose that is bounded. Then, for all vectors with nonzero we have\n\nLetting go to zero shows that is continuous at \nMoreover, since the constant does not depend on this shows that in fact is uniformly continuous, and even Lipschitz continuous.\n\nConversely, it follows from the continuity at the zero vector that there exists a such that for all vectors with \nThus, for all non-zero one has\n\nThis proves that is bounded. Q.E.D."@en . . . . . "\u041E\u0433\u0440\u0430\u043D\u0438\u0447\u0435\u043D\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440"@ru . . . . . . . . . "Bounded operator"@en . . . . . . . . . . . . . . . . . "\u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043E\u0433\u0440\u0430\u043D\u0438\u0447\u0435\u043D\u043D\u044B\u043C, \u0435\u0441\u043B\u0438 \u043A\u0430\u0436\u0434\u043E\u0435 \u043E\u0433\u0440\u0430\u043D\u0438\u0447\u0435\u043D\u043D\u043E\u0435 \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u043E \u0438\u0441\u0445\u043E\u0434\u043D\u043E\u0433\u043E \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0447\u0435\u0441\u043A\u043E\u0433\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u043E\u043D \u043F\u0435\u0440\u0435\u0432\u043E\u0434\u0438\u0442 \u0432 \u043E\u0433\u0440\u0430\u043D\u0438\u0447\u0435\u043D\u043D\u043E\u0435 \u043C\u043D\u043E\u0436\u0435\u0441\u0442\u0432\u043E \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0447\u0435\u0441\u043A\u043E\u0433\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 . \u041F\u0440\u0438\u0432\u0435\u0434\u0451\u043D\u043D\u043E\u0435 \u0432\u044B\u0448\u0435 \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u0438\u0435 \u043E\u0442\u043D\u043E\u0441\u0438\u0442\u0441\u044F \u043A\u0430\u043A \u043A \u043B\u0438\u043D\u0435\u0439\u043D\u044B\u043C, \u0442\u0430\u043A \u0438 \u043A ."@ru . "Em matem\u00E1tica e, em especial, em an\u00E1lise funcional um operador linear limitado \u00E9 um operador linear L entre espa\u00E7os normados em que a norma de um vetor x est\u00E1 limitado (em sentido a ser definido precisamente abaixo) pela norma de x."@pt . . . . . . "In functional analysis and operator theory, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces (a special type of TVS), then is bounded if and only if there exists some such that for all The smallest such is called the operator norm of and denoted by A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces."@en . "En math\u00E9matiques, la notion d'op\u00E9rateur born\u00E9 est un concept d'analyse fonctionnelle. Il s'agit d'une application lin\u00E9aire L entre deux espaces vectoriels norm\u00E9s X et Y telle que l'image de la boule unit\u00E9 de X est une partie born\u00E9e de Y. On montre qu'ils s'identifient aux applications lin\u00E9aires continues de X dans Y. L'ensemble des op\u00E9rateurs born\u00E9s est muni d'une norme issue des normes de X et de Y, la norme d'op\u00E9rateur."@fr . "\u5728\u6CDB\u51FD\u5206\u6790\u6B64\u4E00\u6578\u5B78\u5206\u652F\u88E1\uFF0C\u6709\u754C\u7DDA\u6027\u7B97\u5B50\u662F\u6307\u5728\u8CE6\u7BC4\u5411\u91CF\u7A7A\u9593X \u53CAY \u4E4B\u9593\u7684\u4E00\u7A2E\u7DDA\u6027\u8B8A\u63DBL\uFF0C\u4F7F\u5F97\u5C0D\u6240\u6709X \u5167\u7684\u975E\u96F6\u5411\u91CFv\uFF0CL(v) \u7684\u7BC4\u6578\u8207v \u7684\u7BC4\u6578\u9593\u7684\u6BD4\u503C\u6703\u4FB7\u9650\u5728\u76F8\u540C\u7684\u6578\u5B57\u5167\u3002\u4EA6\u5373\uFF0C\u5B58\u5728\u4E00\u4E9BM > 0\uFF0C\u4F7F\u5F97\u5C0D\u6240\u6709\u5728X \u5167\u7684v\uFF0C \u5176\u4E2D\u6700\u5C0F\u7684M \u7A31\u70BAL \u7684\u7B97\u5B50\u8303\u6570\u3002\u3002 \u6709\u754C\u7DDA\u6027\u7B97\u5B50\u4E00\u822C\u4E0D\u6703\u662F\u6709\u754C\u51FD\u6578\uFF1B\u5F8C\u8005\u9700\u8981\u5C0D\u6240\u6709\u7684v\uFF0CL(v)\u7684\u7BC4\u6578\u662F\u6709\u754C\u7684\uFF0C\u4F46\u9019\u53EA\u6709\u5728Y \u70BA\u96F6\u5411\u91CF\u7A7A\u9593\u6642\u624D\u6709\u53EF\u80FD\u3002\u7136\u800C\uFF0C\u6709\u754C\u7DDA\u6027\u7B97\u7B26\u70BA\u3002 \u4E00\u500B\u7DDA\u6027\u7B97\u5B50\u70BA\u6709\u754C\u7684\uFF0C\u82E5\u4E14\u552F\u82E5\u5176\u70BA\u9023\u7E8C\u7684\u3002\u56E0\u6B64\u6709\u754C\u7EBF\u6027\u7B97\u5B50\u4E5F\u88AB\u79F0\u4E3A\u8FDE\u7EED\u7EBF\u6027\u7B97\u5B50\u3002"@zh . . "Operador linear limitado"@pt . . . . . . . . . . . "In analisi funzionale un operatore limitato \u00E8 un operatore tra due spazi metrici e tale per cui, comunque si scelga un sottoinsieme limitato , l'insieme \u00E8 un sottoinsieme limitato di . Un operatore lineare continuo limitato tra spazi vettoriali normati \u00E8 una funzione tale per cui il rapporto tra la norma dell'immagine di un vettore e la norma del vettore stesso sia limitato dallo stesso numero per ogni vettore non nullo del dominio. In particolare, un operatore lineare \u00E8 limitato se e solo se \u00E8 continuo."@it . . . "p/b017420"@en . . "\uD568\uC218\uD574\uC11D\uD559\uC5D0\uC11C \uC720\uACC4 \uC791\uC6A9\uC18C(\u6709\u754C\u4F5C\u7528\u7D20, \uC601\uC5B4: bounded operator)\uB294 \uC720\uACC4 \uC9D1\uD569\uC744 \uD56D\uC0C1 \uC720\uACC4 \uC9D1\uD569\uC5D0 \uB300\uC751\uC2DC\uD0A4\uB294, \uB450 \uC704\uC0C1 \uBCA1\uD130 \uACF5\uAC04 \uC0AC\uC774\uC758 \uC120\uD615 \uBCC0\uD658\uC774\uB2E4. \uB450 \uB178\uB984 \uACF5\uAC04 \uC0AC\uC774\uC758 \uACBD\uC6B0, \uC720\uACC4 \uC791\uC6A9\uC18C\uC758 \uAC1C\uB150\uC740 \uC5F0\uC18D \uC120\uD615 \uBCC0\uD658\uC758 \uAC1C\uB150\uACFC \uC77C\uCE58\uD55C\uB2E4."@ko . . . . . . . . . . . "\u95A2\u6570\u89E3\u6790\u5B66\u306B\u304A\u3044\u3066\u6709\u754C\uFF08\u7DDA\u5F62\uFF09\u4F5C\u7528\u7D20\uFF08\u3086\u3046\u304B\u3044\u3055\u3088\u3046\u305D\u3001\u82F1: Bounded\u3008linear\u3009operator\uFF09\u3068\u306F\u3001\u4E8C\u3064\u306E\u30CE\u30EB\u30E0\u7A7A\u9593 X \u304A\u3088\u3073 Y \u306E\u9593\u306E\u7DDA\u578B\u4F5C\u7528\u7D20 L \u3067\u3042\u3063\u3066\u3001X \u306B\u542B\u307E\u308C\u308B\u30BC\u30ED\u3067\u306A\u3044\u3059\u3079\u3066\u306E\u30D9\u30AF\u30C8\u30EB v \u306B\u5BFE\u3057\u3066 L(v) \u306E\u30CE\u30EB\u30E0\u3068 v \u306E\u30CE\u30EB\u30E0\u306E\u6BD4\u304C\u3001v \u306B\u4F9D\u5B58\u3057\u306A\u30441\u3064\u306E\u6570\u306B\u3088\u3063\u3066\u4E0A\u304B\u3089\u8A55\u4FA1\u3055\u308C\u308B\u3088\u3046\u306A\u3082\u306E\u306E\u3053\u3068\u3092\u8A00\u3046\u3002\u8A00\u3044\u63DB\u3048\u308B\u3068\u3001\u6B21\u3092\u6E80\u305F\u3059\u7DDA\u578B\u4F5C\u7528\u7D20 L \u306E\u3053\u3068\u3092\u3001\u6709\u754C\u4F5C\u7528\u7D20\u3068\u8A00\u3046: \u3053\u3053\u3067 \u306F X \u304C\u5099\u3048\u308B\u30CE\u30EB\u30E0\u3067\u3042\u308B\uFF08 \u3082\u540C\u69D8\uFF09\uFF0E\u4E0A\u8A18\u306E\u6B63\u5B9A\u6570 M \u306E\u4E0B\u9650\u306F L \u306E\u4F5C\u7528\u7D20\u30CE\u30EB\u30E0\u3068\u547C\u3070\u308C\u3001 \u3068\u8A18\u8FF0\u3055\u308C\u308B\u3002 X \u304B\u3089 Y \u3078\u306E\u6709\u754C\u4F5C\u7528\u7D20\u5168\u4F53\u306E\u96C6\u5408\u3092 \u3068\u3057\u3066\uFF0C\u306B\u5BFE\u3057\u3066 \u306B\u3088\u3063\u3066\u4F5C\u7528\u7D20\u30CE\u30EB\u30E0\u3092\u8868\u3059\u3053\u3068\u3082\u3042\u308B\uFF0E \u4E00\u822C\u7684\u306B\u3001\u6709\u754C\u4F5C\u7528\u7D20\u306F\u6709\u754C\u95A2\u6570\u3067\u306F\u306A\u3044\u3002\u5F8C\u8005\u306F\u3001\u3059\u3079\u3066\u306E v \u306B\u5BFE\u3057 L(v) \u306E\u30CE\u30EB\u30E0\u304C\u4E0A\u304B\u3089\u8A55\u4FA1\u3055\u308C\u3066\u3044\u308B\u5FC5\u8981\u304C\u3042\u308B\u304C\u3001\u3053\u308C\u306F L \u304C\u96F6\u4F5C\u7528\u7D20\u3067\u306A\u3044\u3068\u8D77\u3053\u308A\u5F97\u306A\u3044\u3002\u6709\u754C\u4F5C\u7528\u7D20\u306F\u3067\u3042\u308B\u3002 \u7DDA\u5F62\u4F5C\u7528\u7D20\u304C\u6709\u754C\u3067\u3042\u308B\u3053\u3068\u3068\u3001\u9023\u7D9A\u3067\u3042\u308B\u3053\u3068\u306F\u5FC5\u8981\u5341\u5206\u3067\u3042\u308B\u3002"@ja . "\u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u043C\u0456\u0436 \u0434\u0432\u043E\u043C\u0430 \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0447\u043D\u0438\u043C\u0438 \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u0438\u043C\u0438 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0430\u043C\u0438 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043E\u0431\u043C\u0435\u0436\u0435\u043D\u0438\u043C, \u044F\u043A\u0449\u043E \u043A\u043E\u0436\u043D\u0443 \u043E\u0431\u043C\u0435\u0436\u0435\u043D\u0443 \u043C\u043D\u043E\u0436\u0438\u043D\u0443 \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0447\u043D\u043E\u0433\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 \u0432\u0456\u043D \u043F\u0435\u0440\u0435\u0432\u043E\u0434\u0438\u0442\u044C \u0432 \u043E\u0431\u043C\u0435\u0436\u0435\u043D\u0443 \u043C\u043D\u043E\u0436\u0438\u043D\u0443 \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0447\u043D\u043E\u0433\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 . \u0414\u0430\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043C\u043E\u0436\u043D\u0430 \u0437\u0430\u0441\u0442\u043E\u0441\u043E\u0432\u0443\u0432\u0430\u0442\u0438 \u0434\u043E \u043B\u0456\u043D\u0456\u0439\u043D\u0438\u0445 \u0456 \u043D\u0435\u043B\u0456\u043D\u0456\u0439\u043D\u0438\u0445 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u0456\u0432. \u0411\u0443\u0434\u044C-\u044F\u043A\u0438\u0439 \u043D\u0435\u043F\u0435\u0440\u0435\u0440\u0432\u043D\u0438\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u0454 \u043E\u0431\u043C\u0435\u0436\u0435\u043D\u0438\u043C."@uk . . . "In de wiskunde is een begrensde operator een lineaire afbeelding tussen genormeerde vectorruimten waarvan de operatornorm eindig is. Onder een begrensde operator is het beeld van een begrensde verzameling weer begrensd. Voor lineaire operatoren is begrensdheid equivalent met continu\u00EFteit."@nl . . . . . . . . . . "Begrensde operator"@nl . . . "\uC720\uACC4 \uC791\uC6A9\uC18C"@ko . . . . . . "In functional analysis and operator theory, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces (a special type of TVS), then is bounded if and only if there exists some such that for all The smallest such is called the operator norm of and denoted by A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function is called \"bounded\" then this usually means that its image is a bounded subset of its codomain. A linear map has this property if and only if it is identically Consequently, in functional analysis, when a linear operator is called \"bounded\" then it is never meant in this abstract sense (of having a bounded image)."@en . . "En math\u00E9matiques, la notion d'op\u00E9rateur born\u00E9 est un concept d'analyse fonctionnelle. Il s'agit d'une application lin\u00E9aire L entre deux espaces vectoriels norm\u00E9s X et Y telle que l'image de la boule unit\u00E9 de X est une partie born\u00E9e de Y. On montre qu'ils s'identifient aux applications lin\u00E9aires continues de X dans Y. L'ensemble des op\u00E9rateurs born\u00E9s est muni d'une norme issue des normes de X et de Y, la norme d'op\u00E9rateur."@fr . . . . . . . . . . . "Un operador lineal acotado u operador acotado es una aplicaci\u00F3n lineal definida sobre un espacio vectorial normado tal que la norma de sus valores puede acotarse. M\u00E1s precisamente, la aplicaci\u00F3n lineal es un operador acotado si y solo s\u00ED:"@es . . . . . "In der Mathematik werden lineare Abbildungen zwischen normierten Vektorr\u00E4umen als beschr\u00E4nkte (lineare) Operatoren bezeichnet, wenn ihre Operatornorm endlich ist. Lineare Operatoren sind genau dann beschr\u00E4nkt, wenn sie stetig sind, weshalb beschr\u00E4nkte lineare Operatoren oft als stetige (lineare) Operatoren bezeichnet werden."@de . . . "455961"^^ . . . . . . "\u6709\u754C\u4F5C\u7528\u7D20"@ja . "In der Mathematik werden lineare Abbildungen zwischen normierten Vektorr\u00E4umen als beschr\u00E4nkte (lineare) Operatoren bezeichnet, wenn ihre Operatornorm endlich ist. Lineare Operatoren sind genau dann beschr\u00E4nkt, wenn sie stetig sind, weshalb beschr\u00E4nkte lineare Operatoren oft als stetige (lineare) Operatoren bezeichnet werden."@de . . "\u5728\u6CDB\u51FD\u5206\u6790\u6B64\u4E00\u6578\u5B78\u5206\u652F\u88E1\uFF0C\u6709\u754C\u7DDA\u6027\u7B97\u5B50\u662F\u6307\u5728\u8CE6\u7BC4\u5411\u91CF\u7A7A\u9593X \u53CAY \u4E4B\u9593\u7684\u4E00\u7A2E\u7DDA\u6027\u8B8A\u63DBL\uFF0C\u4F7F\u5F97\u5C0D\u6240\u6709X \u5167\u7684\u975E\u96F6\u5411\u91CFv\uFF0CL(v) \u7684\u7BC4\u6578\u8207v \u7684\u7BC4\u6578\u9593\u7684\u6BD4\u503C\u6703\u4FB7\u9650\u5728\u76F8\u540C\u7684\u6578\u5B57\u5167\u3002\u4EA6\u5373\uFF0C\u5B58\u5728\u4E00\u4E9BM > 0\uFF0C\u4F7F\u5F97\u5C0D\u6240\u6709\u5728X \u5167\u7684v\uFF0C \u5176\u4E2D\u6700\u5C0F\u7684M \u7A31\u70BAL \u7684\u7B97\u5B50\u8303\u6570\u3002\u3002 \u6709\u754C\u7DDA\u6027\u7B97\u5B50\u4E00\u822C\u4E0D\u6703\u662F\u6709\u754C\u51FD\u6578\uFF1B\u5F8C\u8005\u9700\u8981\u5C0D\u6240\u6709\u7684v\uFF0CL(v)\u7684\u7BC4\u6578\u662F\u6709\u754C\u7684\uFF0C\u4F46\u9019\u53EA\u6709\u5728Y \u70BA\u96F6\u5411\u91CF\u7A7A\u9593\u6642\u624D\u6709\u53EF\u80FD\u3002\u7136\u800C\uFF0C\u6709\u754C\u7DDA\u6027\u7B97\u7B26\u70BA\u3002 \u4E00\u500B\u7DDA\u6027\u7B97\u5B50\u70BA\u6709\u754C\u7684\uFF0C\u82E5\u4E14\u552F\u82E5\u5176\u70BA\u9023\u7E8C\u7684\u3002\u56E0\u6B64\u6709\u754C\u7EBF\u6027\u7B97\u5B50\u4E5F\u88AB\u79F0\u4E3A\u8FDE\u7EED\u7EBF\u6027\u7B97\u5B50\u3002"@zh . . . . . . . . . . . "Operador lineal acotado"@es . . . . . . . . . . . "\u041E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u043C\u0456\u0436 \u0434\u0432\u043E\u043C\u0430 \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0447\u043D\u0438\u043C\u0438 \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u0438\u043C\u0438 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0430\u043C\u0438 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043E\u0431\u043C\u0435\u0436\u0435\u043D\u0438\u043C, \u044F\u043A\u0449\u043E \u043A\u043E\u0436\u043D\u0443 \u043E\u0431\u043C\u0435\u0436\u0435\u043D\u0443 \u043C\u043D\u043E\u0436\u0438\u043D\u0443 \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0447\u043D\u043E\u0433\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 \u0432\u0456\u043D \u043F\u0435\u0440\u0435\u0432\u043E\u0434\u0438\u0442\u044C \u0432 \u043E\u0431\u043C\u0435\u0436\u0435\u043D\u0443 \u043C\u043D\u043E\u0436\u0438\u043D\u0443 \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0447\u043D\u043E\u0433\u043E \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u0433\u043E \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0443 . \u0414\u0430\u043D\u0435 \u043E\u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u043C\u043E\u0436\u043D\u0430 \u0437\u0430\u0441\u0442\u043E\u0441\u043E\u0432\u0443\u0432\u0430\u0442\u0438 \u0434\u043E \u043B\u0456\u043D\u0456\u0439\u043D\u0438\u0445 \u0456 \u043D\u0435\u043B\u0456\u043D\u0456\u0439\u043D\u0438\u0445 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440\u0456\u0432. \u0411\u0443\u0434\u044C-\u044F\u043A\u0438\u0439 \u043D\u0435\u043F\u0435\u0440\u0435\u0440\u0432\u043D\u0438\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u0454 \u043E\u0431\u043C\u0435\u0436\u0435\u043D\u0438\u043C."@uk . . . "In de wiskunde is een begrensde operator een lineaire afbeelding tussen genormeerde vectorruimten waarvan de operatornorm eindig is. Onder een begrensde operator is het beeld van een begrensde verzameling weer begrensd. Voor lineaire operatoren is begrensdheid equivalent met continu\u00EFteit."@nl . . . . . . . . . . "15447"^^ . . "Bounded operator"@en . . . . . . . . . "Operator liniowy ograniczony"@pl . . "\u041E\u0431\u043C\u0435\u0436\u0435\u043D\u0438\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440"@uk .