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In the field of calculus of variations in mathematics, the method of Lagrange multipliers on Banach spaces can be used to solve certain infinite-dimensional constrained optimization problems. The method is a generalization of the classical method of Lagrange multipliers as used to find extrema of a function of finitely many variables. ==The Lagrange multiplier theorem for Banach spaces== Let ''X'' and ''Y'' be real Banach spaces. Let ''U'' be an open subset of ''X'' and let ''f'' : ''U'' → R be a continuously differentiable function. Let ''g'' : ''U'' → ''Y'' be another continuously differentiable function, the ''constraint'': the objective is to find the extremal points (maxima or minima) of ''f'' subject to the constraint that ''g'' is zero. Suppose that ''u''0 is a ''constrained extremum'' of ''f'', i.e. an extremum of ''f'' on : Suppose also that the Fréchet derivative D''g''(''u''0) : ''X'' → ''Y'' of ''g'' at ''u''0 is a surjective linear map. Then there exists a Lagrange multiplier ''λ'' : ''Y'' → R in ''Y''∗, the dual space to ''Y'', such that : Since D''f''(''u''0) is an element of the dual space ''X''∗, equation (L) can also be written as : where (D''g''(''u''0))∗(''λ'') is the pullback of ''λ'' by D''g''(''u''0), i.e. the action of the adjoint map (D''g''(''u''0))∗ on ''λ'', as defined by : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lagrange multipliers on Banach spaces」の詳細全文を読む スポンサード リンク
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