Functional Analysis Review

Transcription

1 Outline 9.520: Statistical Learning Theory and Applications February 8, 2010

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3 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all a, b R and v, w, x V: 1 v + w = w + v 2 (v + w) + x = v + (w + x) 3 There exists 0 V such that v + 0 = v for all v V 4 For every v V there exists v V such that v + ( v) = 0 5 a(bv) = (ab)v 6 1v = v 7 (a + b)v = av + bv 8 a(v + w) = av + aw Example: R n, space of polynomials, space of functions.

4 Inner Product Outline An inner product is a function, : V V R such that for all a, b R and v, w, x V: 1 v, w = w, v 2 av + bw, x = a v, x + b w, x 3 v, v 0 and v, v = 0 if and only if v = 0. v, w V are orthogonal if v, w = 0. Given W V, we have V = W W, where W = { v V v, w = 0 for all w W }. Cauchy-Schwarz inequality: v, w v, v 1/2 w, w 1/2.

5 Norm Outline A norm is a function : V R such that for all a R and v, w V: 1 v 0, and v = 0 if and only if v = 0 2 av = a v 3 v + w v + w Can define norm from inner product: v = v, v 1/2.

6 Metric Outline A metric is a function d: V V R such that for all v, w, x V: 1 d(v, w) 0, and d(v, w) = 0 if and only if v = w 2 d(v, w) = d(w, v) 3 d(v, w) d(v, x) + d(x, w) Can define metric from norm: d(v, w) = v w.

7 Basis Outline B = {v 1,..., v n } is a basis of V if every v V can be uniquely decomposed as for some a 1,..., a n R. v = a 1 v a n v n An orthonormal basis is a basis that is orthogonal ( v i, v j = 0 for i j) and normalized ( v i = 1).

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9 Outline Hilbert Space, overview Goal: to understand Hilbert spaces (complete inner product spaces) and to make sense of the expression f = Need to talk about: 1 Cauchy sequence 2 Completeness 3 Density 4 Separability f, φ i φ i, f H i=1

10 Outline Cauchy Sequence Recall: lim n x n = x if for every ɛ > 0 there exists N N such that x x n < ɛ whenever n N. (x n ) n N is a Cauchy sequence if for every ɛ > 0 there exists N N such that x m x n < ɛ whenever m, n N. Every convergent sequence is a Cauchy sequence (why?)

11 Completeness Outline A normed vector space V is complete if every Cauchy sequence converges. Examples: 1 Q is not complete. 2 R is complete (axiom). 3 R n is complete. 4 Every finite dimensional normed vector space (over R) is complete.

12 Hilbert Space Outline A Hilbert space is a complete inner product space. Examples: 1 R n 2 Every finite dimensional inner product space. 3 l 2 = {(a n ) n=1 a n R, n=1 a2 n < } 4 L 2 ([0, 1]) = {f: [0, 1] R 1 0 f(x)2 dx < }

13 Density Outline Y is dense in X if Y = X. Examples: 1 Q is dense in R. 2 Q n is dense in R n. 3 Weierstrass approximation theorem: polynomials are dense in continuous functions (with the supremum norm, on compact domains).

14 Separability Outline X is separable if it has a countable dense subset. Examples: 1 R is separable. 2 R n is separable. 3 l 2, L 2 ([0, 1]) are separable.

15 Outline Orthonormal Basis A Hilbert space has a countable orthonormal basis if and only if it is separable. Can write: Examples: f = f, φ i φ i for all f H. i=1 1 Basis of l 2 is (1, 0,..., ), (0, 1, 0,... ), (0, 0, 1, 0,... ),... 2 Basis of L 2 ([0, 1]) is 1, 2 sin 2πnx, 2 cos 2πnx for n N

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17 Matrix Outline Every linear operator L: R m R n can be represented by an m n matrix A. If A R m n, the transpose of A is A R n m satisfying Ax, y R m = (Ax) y = x A y = x, A y R n for every x R n and y R m. A is symmetric if A = A.

18 Outline Eigenvalues and Eigenvectors Let A R n n. A nonzero vector v R n is an eigenvector of A with corresponding eigenvalue λ R if Av = λv. Symmetric matrices have real eigenvalues. Spectral Theorem: Let A be a symmetric n n matrix. Then there is an orthonormal basis of R n consisting of the eigenvectors of A. Eigendecomposition: A = VΛV, or equivalently, n A = λ i v i v i. i=1

19 Outline Singular Value Decomposition Every A R m n can be written as A = UΣV, where U R m m is orthogonal, Σ R m n is diagonal, and V R n n is orthogonal. Singular system: Av i = σ i u i A u i = σ i v i AA u i = σ 2 i u i A Av i = σ 2 i v i

20 Matrix Norm Outline The spectral norm of A R m n is A spec = σ max (A) = λ max (AA ) = λ max (A A). The Frobenius norm of A R m n is m n A F = a 2 ij = min{m,n} i=1 j=1 i=1 σ 2 i.

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22 Linear Operator Outline An operator L: H 1 H 2 is linear if it preserves the linear structure. A linear operator L: H 1 H 2 is bounded if there exists C > 0 such that Lf H2 C f H1 for all f H 1. A linear operator is continuous if and only if it is bounded.

23 Outline Adjoint and Compactness The adjoint of a bounded linear operator L: H 1 H 2 is a bounded linear operator L : H 2 H 1 satisfying Lf, g H2 = f, L g H1 for all f H 1, g H 2. L is self-adjoint if L = L. Self-adjoint operators have real eigenvalues. A bounded linear operator L: H 1 H 2 is compact if the image of the unit ball in H 1 has compact closure in H 2.

24 Outline Spectral Theorem for Compact Self-Adjoint Operator Let L: H H be a compact self-adjoint operator. Then there exists an orthonormal basis of H consisting of the eigenfunctions of L, Lφ i = λ i φ i and the only possible limit point of λ i as i is 0. Eigendecomposition: L = λ i φ i, φ i. i=1