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Function spaces: notion and notations

Module by: Veronique Delouille. E-mail the author

A Hilbert space is a complete normed space whose norm is indexed by an inner (or scalar) product.

Two disjoint subspaces AA and BB of a space SS form a direct sum decomposition of SS if every element of SS can be written uniquely as a sum of an element of AA and an element of BB. The notation S=A⊕BS=A⊕B is then used.

A measurable function ff belongs to the Lebesgue space Lp(R),1≤p<∞Lp(R),1≤p<∞ if

f p = ∫ - ∞ + ∞ f ( x ) p 1 / p < ∞ . f p = ∫ - ∞ + ∞ f ( x ) p 1 / p < ∞ .
(1)

An example of a Hilbert space is the Lebesgue space L2(R)L2(R) of measurable and square integrable functions. Indeed, the norm ·2·2 is induced by the scalar product

f , g = ∫ f ( x ) g ( x ) ¯ d x , f , g = ∫ f ( x ) g ( x ) ¯ d x ,
(2)

where g(x)¯g(x)¯ denotes the complex conjugate of g(x)g(x). Two functions are said to be orthogonal in L2(R)L2(R) if their inner product is zero.

The Lebesgue measure can be replaced by a more general measure μμ, leading to the weighted space L2(μ)L2(μ), which has as inner product

f , g d μ = ∫ f ( x ) g ( x ) ¯ d μ ( x ) f , g d μ = ∫ f ( x ) g ( x ) ¯ d μ ( x )
(3)

and which contains the functions that have a finite norm fdμ:=f,fdμ<∞fdμ:=f,fdμ<∞.

A countable subset {fk}{fk} of functions belonging to a Hilbert space is a Riesz basis if every element ff of the space can be written uniquely as f=∑kckfkf=∑kckfk, and if positive constants AA and BB exist such that

A f 2 2 ≤ ∑ k c k 2 ≤ B f 2 2 . A f 2 2 ≤ ∑ k c k 2 ≤ B f 2 2 .
(4)

A Riesz basis is an orthogonal basis if the fkfk are mutually orthogonal. In this case, A=B=1A=B=1.

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