2.4 Hilbert Spaces. Outline

Transcription

1 2.4 Hilbert Spaces Tom Lewis Spring Semester 2017 Outline Hilbert spaces L 2 ([a, b]) Orthogonality Approximations

2 Definition A Hilbert space is an inner product space which is complete in the norm defined by its inner product. Example R n, endowed with the dot-product, is a Hilbert space.

3 Theorem The space l 2 is a Hilbert space. Proof. Exercise. Theorem The space C([0, 1]) with the inner product f, g = 1 0 f (x)g(x)dx is not complete in the norm defined by its inner product.

4 Note The natural completion of C([a, b]) in this inner product is called L 2 ([a, b]), the Hilbert space of square integrable functions on [a, b]. A proper study of this space requires the Lebesgue integral, but we will work with it somewhat informally. Definition The elements x and y in V are orthogonal, written x y, provided that x, y = 0.

5 Problem Verify each claim. 1. In R n, the standard unit vectors are orthogonal. 2. The functions sin(x) and cos(x) are orthogonal in L 2 ([ π, π]). Definition Let M be a subspace of an inner product space and define often called M-perp. M = {y : y, x = 0 for all x V },

6 Theorem Let M be a subspace of an inner product space. The set M is a closed subspace. Definition (Convex) A set C in a vector space V is called convex provided that: whenever x, y C and 0 t 1, the vector also lies in C. (1 t)x + ty

7 Theorem (Parallelogram Law) Let V be an inner product space. For all vectors x, y V, 1. x + y 2 + x y 2 = 2 x y 2 and 2. x y 2 = 2 x y 2 4 x + y 2 2. Theorem Every nonempty, closed, convex set E in a Hilbert space contains a unique element of smallest norm. Proof. Exercise.

8 Theorem (Projection Theorem) Let M be a closed subspace of a Hilbert space H. 1. Every x H has a unique decomposition x = Px + Qx into a sum where Px M and Qx M. 2. Px and Qx are the nearest points to x in M and M, respectively. 3. The mappings P : H M and Q : H M are linear. 4. x 2 = Px 2 + Qx 2. Definition A set of vectors {u α : α I } is called orthonormal provided that u α, u β = { 1 if α = β 0 if α β. In other words, each vector has length 1 and each pair of vectors are orthogonal.

9 Problem Show that the standard unit vectors in R n form an orthonormal set. Problem For each pair of nonnegative integers n and i, let x ni = { 1 if i = n 0 if i n Let x(n) = {x ni : i 1}. Show that the collection {x(n)} is an orthonormal set in l 2.

10 Problem Show that the collection of functions 1, cos(x), sin(x) 2π π, cos(2x) π π, sin(2x) π,... is an orthonormal set in L 2 ([ π, π]). (Exercise) Theorem Let V be a Hilbert space. Let x V, let {u i : 1 i n} be a finite orthonormal set in V, and let M = span{u 1, u 2,..., u n }. Then M is a closed subspace of V and the unique closest element to x in M is given by Px = n x, u i u i. (1) i=1

11 Definition The numbers { x, u i : 1 i n} in Equation (1) are called the Fourier coefficients of x relative to the orthonormal basis for M. Example Let f (x) = x π if π x < π/2 x if π/2 x < π/2 π x if π/2 x π.

12 Figure: The graph of the function f. The subspace Let M be the span of the elements 1, cos(x), sin(x) 2π π, cos(2x) π π, sin(2x) π,..., cos(10x), sin(10x) π π

13 The projection The projection of f onto M is given by Pf (x) = 4 sin(x) π + 4 sin(3x) 9π 4 sin(5x) 25π + 4 sin(7x) 49π 4 sin(9x) 81π Figure: The graph of the projection, Pf.