Metric and Hilbert Spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 4 November 2014

Lecture 33: Inner product spaces and orthogonality

An inner product space is a vector space V over with a function V×V (v1,v2) v1,v2 such that

(a) If v1,v2,v3V and c1,c2 then c1v1+c2v2,v3= c1v1,v3+ c2v2,v3.
(b) If v1,v2,v3V and c1,c2 then v3,c1v1+c2v2= c1v3,v1+ c2v3,v2.
(c) If v1,v2V then v2,v1=v1,v2.
(d) If vV and v,v=0 then v=0.
(e) If vV then v,v0.

Let (V,) be an inner product space. Define :V0 by v=v,v.

HW: Show that (V,·) is a normed vector space.

A Hilbert space is an inner product space (V,) such that V is a complete metric space.

Orthogonal complements

Let (V,) be an inner product space and let WV be a subspace of V.

The orthogonal complement of W in V is W= { vV|ifwW thenv,w =0 } .

If V is a Hilbert space and W is closed then V=WW.

Let V be an inner product space and let W be a subspace of V. An orthogonal projection onto W is a linear transformation P:VV such that

(a) if vV then P(v)W,
(b) if vW then v-P(v)W.

(sub-Theorem) Let V be a Hilbert space and let W be a subspace of V. There exists an orthogonal projection P:VV onto W if and only if W is closed.

Orthogonality

HW: Let (V,) be an inner product space. Show that d: V V* w φw: V v v,w is a linear transformation. Let W be a subspace of V. Show that W=wW ker(φw).

Let V be a Hilbert space.

An orthonormal sequence in V is a sequence a1,a2,a3, in V such that if i,j>0 then ai,aj= { 0, ifij, 1, ifi=j.

HW: Let a1,a2, be an orthonormal sequence in V. Let W=span{a1,a2,}. Show that

(a) If vV then n0 v,an2 v2. (Bessel's inequality)
(b) P:VV given by P(v)=n>0 v,anan is an orthogonal projection onto W.
(c) If W=V then {a1,a2,a3,} is a Schauder basis of V i.e., every vV can be written uniquely as v=λ1a1+λ2a2+.

HW: (Fourier analysis) Let e0,e1,e-1,e2,e-2, in L2[0,2π] be given by em(t)=12π eimt. Show that e0,e1,e-1,e2,e-2, is an orthonormal basis of L2[0,2π].

Gram-Schmidt

Let v1,v2, be a sequence of linearly independent vectors in V. Define a1=v1v1and an+1= vn+1- vn+1,a1a1- - vn+1,anan vn+1- vn+1,a1a1- - vn+1,anan . Then a1,a2, is an orthonormal sequence in V.

Notes and References

These are a typed copy of Lecture 33 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on September 23, 2014.

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