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Hilbert Spaces Chapter 3 Hilbert spaces Contents Introduction .......................................... ............. 3.2 Innerproducts....................................... ............... 3.2 Cauchy-Schwarzinequality . ................ 3.2 Inducednorm ........................................ .............. 3.3 Hilbertspace......................................... .............. 3.4 Minimum norm optimization problems . ............. 3.5 Chebyshevsets ....................................... .............. 3.5 Projectors ........................................... ............. 3.6 Orthogonality .......................................... ............ 3.6 Orthogonalcomplements ................................. ............... 3.11 Orthogonalprojection ................................... ............... 3.12 Directsum........................................... ............. 3.13 Orthogonalsets ....................................... .............. 3.14 Gram-Schmidtprocedure . ................. 3.15 Approximation......................................... ............. 3.17 Normalequations ...................................... .............. 3.17 Grammatrices......................................... ............. 3.18 Orthogonalbases ..................................... ............... 3.18 Approximation and Fourier series . ................. 3.18 Linearvarieties ........................................ ............. 3.20 Dualapproximationproblem ............................... ............... 3.20 Applications........................................... ............ 3.22 Fourierseries ........................................ .............. 3.24 Complete orthonormal sequences / countable orthonormal bases . ......................... 3.26 Wavelets............................................. ............ 3.26 Vectorspaces ........................................ .............. 3.27 Normedspaces....................................... ............... 3.27 Innerproductspaces................................. .................. 3.28 Summary ............................................ ............ 3.28 Minimum distance to a convex set . ............... 3.29 Projection onto convex sets . ............... 3.30 Summary ............................................ ............ 3.31 3.1 3.2 c J. Fessler, November 5, 2004, 17:8 (student version) Key missing geometrical concepts: angle and orthogonality (“right angles”). 3.1 Introduction We now turn to the subset of normed spaces called Hilbert spaces, which must have an inner product. These are particularly useful spaces in applications/analysis. Why not introduce Hilbert first then? For generality: it is helpful to see which properties are general to vector spaces, or to normed spaces, vs which require additional assumptions like an inner product. Overview inner product • orthogonality • orthogonal projections • applications • least-squares minimization • orthonormalization of a basis • Fourier series • General forms of things you have seen before: Cauchy-Schwarz, Gram-Schmidt, Parseval’s theorem 3.2 Inner products Definition. A pre-Hilbert space, aka an inner product space , is a vector space defined on the field = R or = C, along with an inner product operation , : , which must satisfy the followingX axioms x, y F, α .F 1. x, y = y, x ∗ (Hermitianh· symmetry·i X×X→F), where ∗ denotes complex conjugate. ∀ ∈X ∈F 2. hx + yi , zh = xi , z + y, z (additivity) 3. hαx, y =i α xh, y i(scalingh )i 4. hx, x i 0 andh x,i x = 0 iff x = 0. (positive definite) h i ≥ h i Properties of inner products Bilinearity property: α x , β y = α β∗ x , y . i i j j i j h i j i * i j + i j X X X X Lemma. In an inner product space, if x, y = 0 for all y, then x = 0. Proof. Let y = x. h i 2 Cauchy-Schwarz inequality Lemma. For all x,y in an inner product space, x, y x, x y, y = x y (see induced norm below), |h i| ≤ h i h i k k k k with equality iff x and y are linearly dependent.p p Proof. For any λ the positive definiteness of , ensures that ∈F h· ·i 0 x λy, x λy = x, x λ y, x λ∗ x, y + λ 2 y, y . ≤ h − − i h i − h i − h i | | h i If y = 0, the inequality holds trivially. Otherwise, consider λ = x, y / y, y and we have h i h i 0 x, x y, x 2 / y, y . ≤ h i−|h i| h i Rearranging yields y, x x, x y, y = x y . The proof about equality|h conditionsi| ≤ h is Problemih i 3.1k. k k k 2 p This result generalizes all the “Cauchy-Schwarz inequalities” you have seen in previous classes, e.g., vectors in Rn, random variables, discrete-time and continuous-time signals, each of which corresponds to a particular inner product space. c J. Fessler, November 5, 2004, 17:8 (student version) 3.3 Angle Thanks to this inequality, we can generalize the notion of the angle between vectors to any general inner product space as follows: x, y θ = cos−1 |h i| , x, y = 0. x y ∀ 6 k k k k This definition is legitimate since the argument of cos−1 will always be between 0 and 1 due to the Cauchy-Schwarz inequality. Induced norm Proposition. In an inner product space ( , , ), the induced norm x = x, x is indeed a norm. X h· ·i k k h i Proof. What must we show? p The first axiom ensures that x, x is real. • x 0 with equality iff x =h 0 followsi from Axiom 4. • kαxk ≥= αx, αx = α x, αx = α αx, x ∗ = αα∗ x, x ∗ = α x, x = α x , using Axioms 1 and 3. • k k h i h i h i h i | | h i | | k k The only condition remaining to be verified is the triangle inequality: x + y 2 = x, x + x, y + y, x + y, y • p p p p k k p h i h i h i h i = x 2 + 2 real( x, y ) + y 2 x 2 + 2 x, y + y 2 x 2 + 2 x y + y 2 = ( x + y )2 . 2 (Recallk k if z = a +h ıb, theni ak=k real(≤ kz) k √a2|h+ b2 =i| z k.) k ≤ k k k k k k k k k k k k ...................................................≤ ...................................................| | ................ Any inner product space is necessarily a normed space. Is the reverse true? Not in general. The following property distinguishes inner product spaces from mere normed spaces. Lemma. (The parallelogram law.) In an inner product space: x + y 2 + x y 2 = 2 x 2 + 2 y 2 , x, y . (3-1) k k k − k k k k k ∀ ∈X Proof. Expand the norms into inner products and simplify. 2 x x-y x+y y Remarkably, the converse of this Lemma also holds (see, e.g., problem [2, p. 175]). Proposition. If ( , ) is a normed space over C or R, and its norm satisfies the parallelogram law (3-1), then is also an inner product space,X withk·k inner product: X 1 x, y = x + y 2 x y 2 + i x + iy 2 i x iy 2 . h i 4 k k − k − k k k − k − k Proof. homework challenge problem. Continuity of inner products Lemma. In an inner product space ( , , ), if xn x and yn y, then xn, yn x, y . Proof. X h· ·i → → h i → h i x , y x, y = x , y x, y + x, y x, y x , y x, y + x, y x, y |h n ni − h i| |h n ni − h ni h ni − h i| ≤ |h n ni − h ni| |h ni − h i| = x x, y + x, y y |h n − ni| |h n − i| x x y + x y y by Cauchy-Schwarz ≤ k n − k k nk k k k n − k x x M + x y y since y is convergent and hence bounded ≤ k n − k k k k n − k n 0 as n . → → ∞ Thus x , y x, y . 2 h n ni → h i 3.4 c J. Fessler, November 5, 2004, 17:8 (student version) Examples Many of the normed spaces we considered previously are actually induced by suitable inner product space. Example. In Euclidean space, the usual inner product (aka “dot product”) is n x, y = a b , where x = (a ,...,a ) and y = (b ,...,b ). h i i i 1 n 1 n i=1 X Verifying the axioms is trivial. The induced norm is the usual `2 norm. Example. For the space ` over the complex field, the usual inner product is1 x, y = a b∗. 2 h i i i i The Holder¨ inequality, which is equivalent to the Cauchy-Schwarz inequality for this space, ensures that x, y x 2 y 2 . So the inner product is indeed finite for x, y ` . Thus ` is not only a Banach space, itP is also an inner|h producti| space. ≤ k k k k ∈ 2 2 Example. What about ` for p = 2? Do suitable inner products exist? p 6 Consider = R2, with x = (1, 0) and y = (0, 1). X k·kp · The parallelogram law holds (for this x and y) iff 2(1 + 1)2/p = 2 12 + 2 12, i.e., iff 22/p = 2. · · Thus `2 is only inner product space in the `p family of normed spaces. Example. The space of measurable functions on [a, b] with inner product b f, g = w(t)f(t)g∗(t) dt, h i Za where w(t) > 0, t is some (real) weighting function. Choosing w = 1 yields [a, b]. ∀ L2 Hilbert space Definition. A complete inner product space is called a Hilbert space. In other words, a Hilbert space is a Banach space along with an inner product that induces its norm. The addition of the inner product opens many analytical doors, as we shall see. The concept “complete” is appropriate here since any inner product space is a normed space. All of the preceding examples of inner product spaces were complete vector spaces (under the induced norm). Example. The following is an inner product space, but not a Hilbert space, since it is incomplete: b R [a, b] = f : [a, b] R : Riemann integral f 2(t) dt < , 2 → ∞ ( Za ) with inner product (easily verified to satisfy the axioms): f, g = b f(t)g(t) dt . h i a 1Note that the conjugate goes with the second argument become of Axiom 3. I haveR heard that some treatments scale the second argument in Axiom 3, which affects where the conjugates go in the inner products. c J. Fessler, November 5, 2004, 17:8 (student version) 3.5 Minimum norm optimization problems Section 3.3 is called “the projection theorem” and it is about a certain type of minimum norm problem. Before focusing on that specific minimum norm problem, we consider the broad family of such problems.
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