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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
Let be an arbitrary set and a Hilbert space of real-valued functions on , equipped with pointwise addition and pointwise scalar multiplication.The evaluation functional over the Hilbert space of functions is a linear functional that evaluates each function at a point ,
The space of all test functions, ... The justification for this common practice is detailed below. ... is even a Hilbert space. [7]
Let denote a random variable with domain and distribution .Given a symmetric, positive-definite kernel: the Moore–Aronszajn theorem asserts the existence of a unique RKHS on (a Hilbert space of functions : equipped with an inner product , and a norm ‖ ‖) for which is a reproducing kernel, i.e., in which the element (,) satisfies the reproducing property
where H(D) is the space of holomorphic functions in D. Then L 2,h (D) is a Hilbert space: it is a closed linear subspace of L 2 (D), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ƒ in D
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 ...
In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operator norm.