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w ([0, 1]) is identical with the Hilbert space L 2 ([0, 1], μ) where the measure μ of a Lebesgue-measurable set A is defined by = (). Weighted L 2 spaces like this are frequently used to study orthogonal polynomials , because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.
As any Hilbert space, every space is linearly isometric to a suitable (), where the cardinality of the set is the cardinality of an arbitrary basis for this particular . If we use complex-valued functions, the space L ∞ {\displaystyle L^{\infty }} is a commutative C*-algebra with pointwise multiplication and conjugation.
However, there are RKHSs in which the norm is an L 2-norm, such as the space of band-limited functions (see the example below). An RKHS is associated with a kernel that reproduces every function in the space in the sense that for every x {\displaystyle x} in the set on which the functions are defined, "evaluation at x {\displaystyle x} " can be ...
The simplest example of a direct integral are the L 2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions (,).
Note that closed and bounded sets are not in general weakly compact in Hilbert spaces (consider the set consisting of an orthonormal basis in an infinite-dimensional Hilbert space which is closed and bounded but not weakly compact since it doesn't contain 0). However, bounded and weakly closed sets are weakly compact so as a consequence every ...
The space K(ℓ 2) of compact operators on the Hilbert space ℓ 2 has a Schauder basis. For every x, y in ℓ 2, let x ⊗ y denote the rank one operator v ∈ ℓ 2 → <v, x > y. If {e n} n ≥ 1 is the standard orthonormal basis of ℓ 2, a basis for K(ℓ 2) is given by the sequence [17]
The space ℓ 2 is the only ℓ p space that is a Hilbert space, since any norm that is induced by an inner product should satisfy the parallelogram law
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