Functional Analysis - Summary

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1 Functional Analysis - Summary 1 ormed spaces 1 Triangle inequalities for R: x + y x + y, a i a i 2 Cauchy s inequality: For a i, b i R, ) 1/2 ) 1/2 a i b i a i 2 b i 2 3 Hölder s inequality: For p, q > 1, with 1/p + 1/q = 1, 4 Minkowski s inequality: For for p > 1: ) 1/p ) 1/q a i b i a i p b i q ) 1/p ) 1/p a i + b i p ) 1/p a i p + b i p 5 Integral versions of the above, eg for Cauchy s inequality: fg f 2 ) 1/2 g 2 ) 1/2 6 A normed space X, ) is a real or complex) vector space X together with a norm, ie a function : X R, such that i) x 0 with x = 0 if and only if x = 0 positivity), ii) λx = λ x for all x X and scalars λ scalar property) iii) x + y x + y for all x, y X triangle inequality) 7 In a normed space: x x n x x n, x y x y reverse triangle) 8 distx, y) = x y defines a metric on X, ie i) distx, y) 0 with equality iff x = y, ii) distx, y) = disty, x), iii) distx, y) distx, z) + distz, y) 9 Examples of normed spaces: R = {x = x 1,, x )} with x = max 1 i x i, or x 1 = 1 x i, or x 2 = 1 x i 2 ) 1/2, or for any p 1, x p = 1 x i p ) 1/p C[0, 1] = {continuous functions on [0, 1]} with f = sup 0 t 1 ft), or f 1 = 1 0 ft) dt, or f 2 = 1 0 ft)2 dt) 1/2 1

2 L 2 [0, 1] = {f : 1 0 ft)2 dt < } with f 2 = 1 0 ft)2 dt) 1/2, or more generally for p 1, L p [0, 1] = {f : 1 0 ft)p dt < } with f p = 1 0 ft)p dt) 1/p ote we identify functions if their difference has norm 0) Sequence spaces l = {bounded sequences x 1, x 2, x 3, )} with x 1, x 2, x 3, ) = sup 1 i< x i For p 1, l p = {x 1, x 2, x 3, ) : 1 x 1 p < } with x 1, x 2, x 3, ) p = 1 x 1 p ) 1/p 10 Convergence of sequences: x n x in X, ) if x n x 0 11 If x n x, y n y, then x n + y n x + y, and λx n λx 12 Continuity: If X, ) is a normed space, f : X R is continuous at x X if whenever x n x we have fx n ) fx) Moreover, f continuous on X if f is continuous at all x X 13 A subset of S of X, ) is open if for all x S there exists δ > 0 such that {y : y x < δ} S 14 A point x is a limit point of S X if there exists a sequence x n S such that x n x A set S is closed if it contains all its limit points 15 A set S X is closed if and only if its complement X \ S is open 16 The closure S of S is the set of all limit points of S A set S is closed iff S = S 17 For all S, S is closed Moreover S is the smallest closed set containing S ie if T is closed and S T then S T ) 2 Completeness and the contraction mapping theorem 1 A sequence x n ) in a normed space X, ) is convergent to x X if given ɛ > 0 there exists n 0 such that x n x < ɛ for all n n 0 Equivalently if x n x 0 as n 2 A sequence x n ) in a normed space X, ) is a Cauchy sequence if given ɛ > 0 there exists n 0 such that x n x m < ɛ for all n, m n 0 3 In a normed space every convergent sequence is a Cauchy sequence 4 A normed space in which every Cauchy sequence is convergent to a point in the space) is called a complete normed space or a Banach space Similarly, a subset Y of a normed space in which every Cauchy sequence is convergent is called complete 5 Every bounded sequence in R has a convergent subsequence 6 The following normed spaces are complete: R, ) This is just the general principle of convergence) R, ), R, 1 ), R, 2 ) C[0, 1], ) L 1 [0, 1], 1 ), L 2 [0, 1], 2 ), L p [0, 1], p ) for all p 1 7 Let Y be a closed subspace or subset of a Banach space X, ) Then Y is complete 2

3 8 Contraction mapping theorem: Let X be a Banach space or a closed subset of a Banach space with norm Let T : X X be a contraction, ie there is a constant 0 < k < 1 such that T x) T y) k x y x, y X) Then T has a unique fixed point, ie there is a unique x X such that T x) = x Moreover, for all y X, we have T n y) x 9 To apply the contraction mapping theorem, write the problem in the form T x = x and show T is a contraction on some complete set, to deduce that the problem has a unique solution T might be a mapping on R for solution of equations of one variable) or on R for solution of equations in variables) or an integral operaator for solution of differential equations) 3 Finite dimensional normed spaces 1 Every bounded sequence in R, ) has a convergent subsequence 2 Two norms A and B on a vector space X are equivalent if there are a, b > 0 such that a x B x A b x B for all x X 3 If A and B are equivalent then x n x in A if and only if x n x in B Similarly, the Cauchy sequences, limit points, closed sets, etc are the same in both norms 4 If X is a finite dimensional vector space then all norms on X are equivalent 5 In a finite dimensional vector space, the convergent sequences, Cauchy sequences, limit points, closed sets, etc are independent of the norm used 4 The space C[0, 1], ) The material of this chapter is stated for C[0, 1], but applies to continuous functions with the supremum norm on any compact subset of a metric or topological space In particular [0, 1] may be replaced with any bounded closed interval [a, b], the interval [0, 2π] with 0 and 2π identified, or the square [0, 1] [0, 1] 1 A vector subspace A of C[0, 1] is an algebra if fg A whenever f A and g A eg C[0, 1], polynomials, trigonometric polynomials, etc are algebras) 2 Stone-Weierstrass Theorem: Let A be a subalgebra of C[0, 1], ) such that a) A contains the constant functions, and b) A separates the points of [0, 1], ie if t, u [0, 1] then we may find f A such that ft) fu) Then A is dense in C[0, 1], that is A = C[0, 1] 3

4 3 Weierstrass Approximation Theorem: The polynomials on [0, 1] are dense in C[0, 1], ) The infinitely differentiable functions on [0, 1] are dense in C[0, 1], ) The two-variable polynomials px, y) on [0, 1] [0, 1] are dense in C[0, 1] [0, 1]), ) The trigonometric polynomials on [0, 2π] with 0 and 2π identified, ie functions of the form n j=1 a j sin jt + m j=0 b j cos jt, are dense in C[0, 2π], ) 4 To prove a result that applies to all continuous functions in, say, C[0, 1], it is often easier to prove it for a dense subset of nice functions such as the polynomials and use density to transfer the result to general continuous functions in C[0, 1] 5 Operators on normed spaces 1 For X, Y normed spaces, we call a mapping T : X Y an operator T is called linear if T λx + µx ) = λt x) + µt x ) for all x, x X and scalars λ, µ), Equivalently T is linear if both T x + x ) = T x) + T x ) and T λx) = λt x) T is continuous if T x n ) T x) whenever x n x T is bounded if there is a number c such that T x c x for all x X) 2 Let T : X Y be linear Then T is continuous if and only if T is bounded 3 BX, Y ) denotes the set of bounded linear operators from X to Y We abbreviate BX, X) to BX) for the operators from X to itself) 4 For T BX, Y ) the induced norm T on T is given by T x T = sup 0 x X x = sup T x, x: x =1 that is the smallest number c such that T x c x for all x X 5 BX, Y ) is a vector space, which becomes a normed space under the induced norm If Y is complete then so is BX, Y ) 6 If T BX, Y ) and U BY, Z) then UT BX, Y ) with UT U T In particular if T BX) then T n T n for positive integers n 7 The identity operator I on X is defined by Ix = x for all x X Thus IT = T I = T if T BX) 8 T BX) is invertble if there is a bounded linear operator T 1 BX) such that T T 1 = I = T 1 T 9 If T, U BX) then T U is invertible if and only if T and U are both invertible, in which case T U) 1 = U 1 T 1 4

5 10 T BX) is invertible if and only if T is surjective and there is a constant c > 0 such that T x c x for all x X 11 Let BX) be a Banach space and let T BX) If T < 1 then T is invertible, with I T ) 1 = I + T + T 2 + which is convergent in BX)) 12 Let BX) be a complex Banach space and let T BX) The spectrum σt ) of T is σt ) = {λ C : T λi is not invertible} 13 If λ C is such that T λi)x = 0 for some x 0 we call λ an eigenvalue of T and x an eigenvector or sometimes an eigenfunction) Eigenvalues all belong to the spectrum 14 Let T BX) where BX) is a Banach space Then a) σt ) is bounded in fact σt ) {λ : λ T }) b) σt ) is a closed subset of C c) σt ) is non-empty 15 Spectral mapping theorem: Let p be a polynomial Then σpt )) = pσt )) 6 Inner product spaces and Hilbert space 1 An inner product space is a vector space X over R or C usually C in what follows) with an inner product < x, y > C or R) such that: i) < λx + µy, w > = λ < x, w > + µ < y, w > x, y, w X; λ, µ scalars) linearity), ii) < x, y >= < y, x > x, y X) symmetry), iii) < x, x > is real, and < x, x > 0 with < x, x >= 0 if and only if x = 0 positivity) 2 From i) and ii) above < w, λx + µy >= λ < w, x > +µ < w, y > x, y, w X; λ, µ scalars) 3 x =< x, x > defines a norm on X If this normed space X, ) is complete ie a Banach space) we call X, < >) a Hilbert space 4 Examples of inner product spaces: R with < x 1,, x ), y 1,, y ) > = x i y i that is the dot product ) is a real) Hilbert space; the norm is x 1,, x ) 2 = x i 2) 1/2 l 2 = {x 1, x 2, ) : x i 2 < } with < x 1, x 2, ), y 1, y 2, ) > = x i y i is a real or complex Hilbert space; the norm is f 2 = x i 2 ) 1/2 C[0, 1] with < f, g >= 1 0 ft)gt)dt is an inner product space but is OT complete; the norm is f 2 = 1 0 ft) 2 dt ) 1/2 L 2 [0, 1] = {f : f is square integrable on [0, 1] with 1 0 ft) 2 dt < } with < f, g >= 1 0 ft)gt)dt is a Hilbert space; the norm is f 2 = 1 0 ft) 2 dt ) 1/2 5

6 Let wt) be a real weight function Then H = {f : 1 0 ft) 2 wt)dt < } with < f, g >= 1 0 ft)gt)wt)dt is a Hilbert space; the norm is f = 1 0 ft) 2 wt) 2 dt ) 1/2 From now on let H be a Hilbert space 5 x and y are orthogonal or perpendicular if < x, y > = 0 6 A set of elements {e i } H is an orthonormal set if < e i, e j > = 0 for all i j and e i = < e i, e i >= 1 for all i 7 The span of a subset {w i } of H is the vector subspace consisting of all finite linear combinations n 1 a i w i 8 If {e i } is orthonormal set with span{e i } dense in H, we call {e i } a complete orthonormal system 9 Schwartz inequality: In an inner product space < x, y > x y 10 Parallelogram law: In an inner product space x + y 2 + x y 2 = 2 x y 2 11 Let x H and let S be a dense subset of H If < x, s >= 0 for all s S then x = 0 In particular, if {e i } is complete and < x, e i >= 0 for all e i then x = 0 12 Let {e i } be a complete orthonormal system ie with span{e i } dense in H) Then if x H, x = < x, e i > e i with this series convergent in H), with x = < x, e i > 2 Parseval s inequality) 13 Gram-Schmidt process: Let x 1, x 2, ) be linearly independent in a Hilbert space H Then we may find an orthonormal sequence e 1, e 2, ) such that for each n we have span{x 1, x 2, } = span{e 1, e 2, } In fact we get e n inductively by e 1 = x 1 / x 1 and ) n n e n+1 = x n+1 < x n+1, e i > e i / x n+1 < x n+1, e i > e i 14 An orthonormal set {e i } is a basis for H if for every x H we have x = a i e i for some a i with the series convergent 15 If for some sequence x i ), span{x 1, x 2, } is dense in H then H has an orthonormal basis 6

7 7 The spectral theorem Throughout this section H is an infinite dimensional complex Hilbert space with a countable basis) 1 An operator T BH) is self-adjoint or Hermitian if < T x, y >=< x, T y > for all x, y H 2 If T is self-adjoint then < T x, x > is real 3 If T is self-adjoint then i) the eigenvalues of T are all real, ii) eigenvectors corresponding to different eigenvalues are orthogonal 4 Let T be self-adjoint Then T = sup x =1 < T x, x > 5 An operator T BH) is compact if for every bounded sequence x n ) the sequence T x n ) has a convergent subsequence Think of a compact operator as one for which the image is nearly finite dimensional ) 6 Let T : L 2 [0, 1] L 2 [0, 1] be such that the following equicontinuity condition holds: for all ɛ > 0 there exists δ > 0 such that T fx) T fy) < ɛ whenever x y < δ for all f L 2 [0, 1] such that f 2 1 Then T is compact 7 If T is compact and r > 0 then there are at most finitely many eigenvalues counted according to multiplicity) such that λ r 8 Let T be a compact self-adjoint operator on H Then either T or T is an eigenvalue of T 9 Spectral theorem for compact self-adjoint operators: Let T be a compact self-adjoint operator on H Then there is an orthonormal sequence e n ) of eigenvectors in H, with T e n = λ n e n, say, with λ n 0 and λ n 0 Then if x H we may write x = a n e n + y, n=1 where n=1 a n 2 <, and y H satisfies < y, e n >= 0 for all n and T y = 0 In particular T x = a n λ n e n n=1 10 Let T be a compact self-adjoint operator on H Then H has an orthonormal basis of eignevectors of T 7