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Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space.
The first three functions in the sequence () = on [,].As converges weakly to =.. The Hilbert space [,] is the space of the square-integrable functions on the interval [,] equipped with the inner product defined by
Thus, unitary operators are just automorphisms of Hilbert spaces, i.e., they preserve the structure (the vector space structure, the inner product, and hence the topology) of the space on which they act. The group of all unitary operators from a given Hilbert space H to itself is sometimes referred to as the Hilbert group of H, denoted Hilb(H ...
For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data.
The vector space of all continuous antilinear functions on H is called the anti-dual space or complex conjugate dual space of H and is denoted by ¯ ′ (in contrast, the continuous dual space of H is denoted by ′), which we make into a normed space by endowing it with the canonical norm (defined in the same way as the canonical norm on the ...
One relies on the symmetric assumption, therefore the inner product structure, in proving the theorem. Also crucial is the fact that the given operator A is defined everywhere (and, in turn, the completeness of Hilbert spaces). The Hellinger–Toeplitz theorem reveals certain technical difficulties in the mathematical formulation of quantum ...
The Hilbert transform can be understood in terms of a pair of functions f(x) and g(x) such that the function = + is the boundary value of a holomorphic function F(z) in the upper half-plane. [32] Under these circumstances, if f and g are sufficiently integrable, then one is the Hilbert transform of the other.
It is essentially the same universal property shared by all definitions of tensor products, irrespective of the spaces being tensored: this implies that any space with a tensor product is a symmetric monoidal category, and Hilbert spaces are a particular example thereof.