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In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding adjoint operator. Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a complex Hilbert space as "generalized complex numbers", then the adjoint of an operator plays the role of the complex conjugate of a complex number. The adjoint of an operator may also be called the Hermitian adjoint, Hermitian conjugate or Hermitian transpose (after Charles Hermite) of and is denoted by or (the latter especially when used in conjunction with the bra–ket notation). ==Definition for bounded operators== Suppose is a complex Hilbert space, with inner product . Consider a continuous linear operator (for linear operators, continuity is equivalent to being a bounded operator). Then the adjoint of is the continuous linear operator satisfying : Existence and uniqueness of this operator follows from the Riesz representation theorem.〔; 〕 This can be seen as a generalization of the ''adjoint'' matrix of a square matrix which has a similar property involving the standard complex inner product. ==Properties== The following properties of the Hermitian adjoint of bounded operators are immediate:〔 # – involutiveness # If is invertible, then so is , with # # , where denotes the complex conjugate of the complex number – antilinearity (together with 3.) # If we define the operator norm of by : then :〔 Moreover, :〔 One says that a norm that satisfies this condition behaves like a "largest value", extrapolating from the case of self-adjoint operators. The set of bounded linear operators on a complex Hilbert space together with the adjoint operation and the operator norm form the prototype of a C *-algebra. 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「Hermitian adjoint」の詳細全文を読む スポンサード リンク
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