Hilbert Spaces 2.11 2002/01/07 2005/09/30 20:17:14.213 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Adan Galvan jago@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu Chaucy complete convergence inner product inner product space orthogonal complement This module introduces Hilbert spaces. Now we consider inner product spaces with nice convergence properties that allow us to define countably-infinite orthonormal bases. A Hilbert space is a complete inner product space. A complete The rational numbers provide an example of an incomplete set. We know that it is possible to construct a sequence of rational numbers which approximate an irrational number arbitrarily closely. It is easy to see that such a sequence will be Cauchy. However, the sequence will not converge to any rational number, and so the rationals cannot be complete. space is one where all Cauchy sequences converge to some vector within the space. For sequence xn to be Cauchy, the distance between its elements must eventually become arbitrarily small: ε ε 0 N ε n m n N ε m N ε x n x m ε For a sequence xn to be convergent to x, the distance between its elements and x must eventually become arbitrarily small: ε ε 0 N ε n n N ε x n x ε Examples are listed below (assuming the usual inner products): V N V N V l 2 (i.e., square summable sequences) V ℒ 2 (i.e., square integrable functions) We will always deal with separable Hilbert spaces, which are those that have a countable A countable set is a set with at most a countably-infinite number of elements. Finite sets are countable, as are any sets whose elements can be organized into an infinite list. Continuums (e.g., intervals of ) are uncountably infinite. orthonormal (ON) basis. A countable orthonormal basis for V is a countable orthonormal set S x k such that every vector in V can be represented as a linear combination of elements in S: y y V α k y k k α k x k Due to the orthonormality of S, the basis coefficients are given by α k x k y We can see this via: x k y x k n i 0 n α i x i n x k i 0 n α i x i n i 0 n α i x k x i αk where δ k i x k x i (where the second equality invokes the continuity of the inner product). In finite n-dimensional spaces (e.g., n or n ), any n-element ON set constitutes an ON basis. In infinite-dimensional spaces, we have the following equivalences: x 0 x 1 x 2 … is an ON basis If x i y 0 for all i, then y 0 y y V y 2 i i x i y 2 (Parseval's theorem) Every y V is a limit of a sequence of vectors in span x 0 x 1 x 2 … Examples of countable ON bases for various Hilbert spaces include: n : e0 … e N - 1 for ei 0 … 0 1 0 … 0 with "1" in the ith position n : same as n l2 : δi n i , for ≔ δi n δ n i (all shifts of the Kronecker sequence) ℒ2 : to be constructed using wavelets ... Say S is a subspace of Hilbert space V. The orthogonal complement of S in V, denoted S⊥ , is the subspace defined by the set x V ⊥ x S . When S is closed, we can write V S S⊥ The orthogonal projection of y onto S, where S is a closed subspace of V, is y ̂ i i x i y x i s.t. x i is an ON basis for S. Orthogonal projection yields the best approximation of y in S: y ̂ argmin x S y x The approximation error ≔ e y y ̂ obeys the orthogonality principle: ⊥ e S We illustrate this concept using V 3 () but stress that the same geometrical interpretation applies to any Hilbert space.

A proof of the orthogonality principle is: ⇔ ⊥ e S i e x i 0 y y ̂ x i 0 y x i y ̂ x i j j x j y x j x i j j x j y x j x i j j y x j δ i − j y x i