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In probability theory, for a probability measure P on a Hilbert space ''H'' with inner product , the covariance of P is the bilinear form Cov: ''H'' × ''H'' → R given by : for all ''x'' and ''y'' in ''H''. The covariance operator ''C'' is then defined by : (from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is self-adjoint (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case). When P is a centred Gaussian measure, ''C'' is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace. Even more generally, for a probability measure P on a Banach space ''B'', the covariance of P is the bilinear form on the algebraic dual ''B''#, defined by : where is now the value of the linear functional ''x'' on the element ''z''. Quite similarly, the covariance function of a function-valued random element (in special cases called random process or random field) ''z'' is : where ''z''(''x'') is now the value of the function ''z'' at the point ''x'', i.e., the value of the linear functional evaluated at ''z''. 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「covariance operator」の詳細全文を読む スポンサード リンク
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