Function spaces: notion and notations 1.2 2008/08/22 15:02:43 GMT-5 2008/09/04 12:07:37.921 GMT-5 Veronique Delouille v.delouille@sidc.be Veronique Delouille v.delouille@sidc.be Daniel Collins Williamson dwilliamson1285@gmail.com A Hilbert space is a complete normed space whose norm is indexed by an inner (or scalar) product. Two disjoint subspaces A and B of a space S form a direct sum decomposition of S if every element of S can be written uniquely as a sum of an element of A and an element of B. The notation S=A⊕B is then used. A measurable function f belongs to the Lebesgue space Lp(R),1≤p<∞ if f p = ∫ - ∞ + ∞ f ( x ) p 1 / p < ∞ . An example of a Hilbert space is the Lebesgue space L2(R) of measurable and square integrable functions. Indeed, the norm ·2 is induced by the scalar product f , g = ∫ f ( x ) g ( x ) ¯ d x , where g(x)¯ denotes the complex conjugate of g(x). Two functions are said to be orthogonal in L2(R) if their inner product is zero. The Lebesgue measure can be replaced by a more general measure μ, leading to the weighted space L2(μ), which has as inner product f , g d μ = ∫ f ( x ) g ( x ) ¯ d μ ( x ) and which contains the functions that have a finite norm fdμ:=f,fdμ<∞. A countable subset {fk} of functions belonging to a Hilbert space is a Riesz basis if every element f of the space can be written uniquely as f=∑kckfk, and if positive constants A and B exist such that A f 2 2 ≤ ∑ k c k 2 ≤ B f 2 2 . A Riesz basis is an orthogonal basis if the fk are mutually orthogonal. In this case, A=B=1.