Linear Normed Spaces (cont.) Inner Product Spaces

Transcription

1 Linear Normed Spaces (cont.) Inner Product Spaces October 6, 017

2 Linear Normed Spaces (cont.) Theorem A normed space is a metric space with metric ρ(x,y) = x y Note: if x n x then x n x, and if {x n} is a Cauchy then so is { x n } Definition (Banach space) A real (complex) normed linear space which is complete is called Banach Lemma Let L X be a linear subspace of a normed space X. Then: 1 If x L then ρ(x,l) = 0 If x / L then ρ(x,l) > 0 Definition (best approximation) If u L s.t. ρ(x,l) = x u then u is called the best approximation of x by elements of L Theorem Let L be a finite dimensional subspace of a normed space X. Then x X u L s.t. ρ(x,l) = x u

3 Linear Dependence and Independence Definition (linear dependence) A collection of vectors {x 1,...,x n} X is called linearly dependent iff n n a 1,...,a n R s.t. a ix i = 0, and a i > 0 Definition (linear independence) Vectors {x 1,...,x n} X are linearly independent iff n a ix i = 0 a 1 =... = a n = 0 Definition (basis of finite-dim space) n linearly independent vectors in R n is called a basis Definition (infinite-dim space) A linear space X is called infinite-dim space iff for any n N there exist n linearly independent vectors

4 Orthogonality Yuliya Gorb Definition (proper subspace) A subspace L X of a metric space (X,ρ) is called a proper subspace of X iff L X Recall: A vector x in the Euclidean space X = R 3 is orthogonal to a plane M iff ρ(x,m) = x in X In normed spaces, in general, there is no concept of orthogonality Question: If L is a proper linear subspace of a normed space X, we still ask if ρ(x,m) = x? NOT TRUE! Theorem (Riesz Lemma) Let L be a proper linear subspace of a normed space X. Then ε > 0 x / L : x = 1 s.t. ρ(x,l) > 1 ε

5 Inner Product Spaces Yuliya Gorb Definition (inner product space) A (linear) normed space X is called an inner product space over a space K (where K {R,C}) iff there is a function (, ) : X X K satisfying the following properties: 1 (x,x) 0 for all x X, and (x,x) = 0 iff x = 0 (positive-definiteness); (x +y,z) = (x,z)+(y,z) for all x,y,z X and (αx,y) = α(x,y) for all x,y X and α K (linearity in the first argument); 3 (x,y) = (y,x) for all x,y X (conjugate symmetry) Properties, 3 imply (x,y +z) = (x,y)+(x,z) for all x,y,z X and (x,αy) = α(x,y) for all x,y X and α K

6 Inner Product Spaces (cont.) Examples: n 1 R n with (x,y) = x i y i and C n with (x,y) = l with (x,y) = x i y i 3 C[a,b] with (x,y) = 4 L [a,b] with (x,y) = b a b 5 W 1, [a,b] =: H 1 [a,b] with (x,y) = b a a x(t)y(t) dx + x(t)y(t) dx x(t)y(t) dx b a x (t)y (t) dx, where x and y are understood in a weak sense n x i y i

7 Inner Product Spaces (cont.) In an inner product space X one can introduce a norm induced by the inner product (, ) via x = (x,x) Theorem (Cauchy-Schwarz inequality) (x,y) x y, x,y X, and equality holds if and only if x and y are linearly dependent Note: x +y = x + y +Re(x,y), x,y X Definition (Hilbert space) Let (X,(, )) be an inner product space over K {R,C}. X is called a Hilbert space, iff is is a complete normed space with respect to the norm induced by the inner product x := (x,x) 1, x X

8 Corollaries of Cauchy-Schwarz inequality Corollary 1: Let (X,(, )) be an inner product space over K. Then x := (x,x) 1 for x X is a norm on X Corollary : Let (X,(, )) be an inner product space, let x := (x,x) 1. For given x,y X the equality x +y = x + y holds iff y = 0 or x = λy for some λ 0 Corollary 3: The inner product (, ) of an inner product space is a K-valued continuous mapping on X X, where the norm topology of X is determined by the inner product Corollary 4: For x (X,(, )): x = sup (x,y) = sup (x, y) y =1 y 1

9 Corollaries of Cauchy-Schwarz inequality (cont.) Theorem (Parallelogram Law) Let X be an inner product space. Then x +y + x y = x + y, x,y X Theorem (Polarization Identity) Let X be an inner product space over K. Then for all x,y X, x +y x y, K = R (x,y) = x +y ( x y x +iy +i x iy ), K = C

10 Corollaries of Cauchy-Schwarz inequality (cont.) Theorem (inner product space characterization) Let (X, ) be a normed space over K. Then if any x,y X, the parallelogram identity holds: x +y + x y = x + y, x,y X, then X is an inner product space with an inner product defined by x +y x y, K = R (x,y) = x +y ( x y x +iy +i x iy ), K = C and (x,x) = x for x X Note: If (X,(, )) is an inner product space the inner product induces a norm on X. We thus have the notions of convergence, completeness, compactness etc.

11 References Yuliya Gorb Hunter/Nachtergaele Applied Analysis pp. 4 5, 8, 91 93, , Naylor/Sell Lineat Operator Theory... pp , 18, 30 3, 7 75