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 36: Proof of Bessel's inequality and Hilbert space projections

Let $H$ be a Hilbert space.
Let $a_1,a_2,a_3,\dots$ be an orthonormal sequence in $H\text{.}$
To show: $\sum _{n\in {\mathbb{Z}}_{>0}}{\mid ⟨x,a_n⟩\mid }^2\le {\Vert x\Vert }^2\text{.}$
To show: $\sum _{n\in {\mathbb{Z}}_{>0}}{⟨x,a_n⟩}^2\le {\Vert x\Vert }^2\text{.}$
To show: $\underset{k\to \infty }{\text{lim}}\left(\sum _{n=1}^k{⟨x,a_n⟩}^2\right)\le {\Vert x\Vert }^2\text{.}$
To show: If $k\in {\mathbb{Z}}_{>0}$ then $\sum _{n=1}^k{⟨x,a_n⟩}^2\le {\Vert x\Vert }^2\text{.}$
Assume $k\in {\mathbb{Z}}_{>0}\text{.}$ Let $$x_k=\sum _{n=1}^k⟨x,a_n⟩a_n$$ so that $${\Vert x_k\Vert }^2=\sum _{n=1}^k{⟨x,a_n⟩}^2\text{.}$$ To show: ${\Vert x_k\Vert }^2\le {\Vert x\Vert }^2\text{.}$
Then $$\begin{array}{rcl}{\displaystyle ⟨x-x_k,x_k⟩}& =& {\displaystyle ⟨x,x_k⟩-⟨x_k,x_k⟩}\\ & =& {\displaystyle \sum _{n=1}^k⟨x,a_n⟩⟨x,a_n⟩-\sum _{n=1}^k⟨x_k,x_k⟩}\\ & =& {\displaystyle 0,}\end{array}$$ and $$\begin{array}{rcl}{\displaystyle {\Vert x\Vert }^2}& =& {\displaystyle ⟨x,x⟩}\\ & =& {\displaystyle ⟨x_k+(x-x_k),x_k+(x-x_k)⟩}\\ & =& {\displaystyle ⟨x_k,x_k⟩+⟨x_k,(x-x_k)⟩+⟨(x-x_k),x_k⟩+⟨x-x_k,x-x_k⟩}\\ & =& {\displaystyle {\Vert x_k\Vert }^2+0+0+{\Vert x-x_k\Vert }^2\text{.}}\end{array}$$ So ${\Vert x_k\Vert }^2\le {\Vert x\Vert }^2\text{.}$

Let $W=\text{span}\{a_1,a_2,\dots \}\text{.}$
To show:	$P:H\to H$ given by $$P\left(x\right)=\sum _{n\in {\mathbb{Z}}_{>0}}⟨x,a_n⟩a_n$$ is an orthogonal projection onto $\bar{W}\text{.}$
To show:	(a) $P:H\to H$ is a function. (b) If $x\in H$ then $P\left(x\right)\in \bar{W}\text{.}$ (c) If $x\in H$ then $x-P\left(x\right)\in {\left(\bar{W}\right)}^{\perp }\text{.}$
(a) To show: If $x\in H$ then $P\left(x\right)=\sum _{n\in {\mathbb{Z}}_{>0}}⟨x,a_n⟩a_n$ exists in $H\text{.}$ Assume $x\in H\text{.}$ Let $$x_k=\sum _{n=1}^k⟨x,a_n⟩a_n\text{.}$$ To show: $\underset{k\to \infty }{\text{lim}}{x}_k$ exists in $H\text{.}$ Since $H$ is complete, we need To show: $x_1,x_2,x_3,\dots$ is a Cauchy sequence in $H\text{.}$ We know: $$\Vert x_k\Vert =\sum _{n=1}^k{⟨x,a_n⟩}^2$$ so that $\Vert x_1\Vert ,\Vert x_2\Vert ,\dots$ is an increasing sequence in ${ℝ}_{\ge 0}$ bounded by $\Vert x\Vert ,$ (by Bessel's inequality). So $\Vert x_1\Vert ,\Vert x_2\Vert ,\dots$ converges. Let $y=\underset{k\to \infty }{\text{lim}}\Vert x_k\Vert \text{.}$ To show: If $\epsilon \in {ℝ}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ such that if $m,n\in {\mathbb{Z}}_{\ge N}$ then $\Vert x_m-x_n\Vert <\epsilon \text{.}$ Assume $\epsilon \in {ℝ}_{>0}\text{.}$ To show: There exists $N\in {\mathbb{Z}}_{>0}$ such that if $m,n\in {\mathbb{Z}}_{\ge N}$ then $\Vert x_m-x_n\Vert <\epsilon \text{.}$ Let $N\in {\mathbb{Z}}_{>0}$ such that if $k\in {\mathbb{Z}}_{\ge 0}$ then $\mid y^2-{\Vert x_k\Vert }^2\mid <\frac{\epsilon }{2}\text{.}$ To show: If $m,n\in {\mathbb{Z}}_{\ge N}$ then $\Vert x_m-x_n\Vert <\epsilon \text{.}$ Assume $m,n\in {\mathbb{Z}}_{\ge N}\text{.}$ To show: $\Vert x_m-x_n\Vert <\epsilon \text{.}$ $$\begin{array}{rcl}{\displaystyle {\Vert x_m-x_n\Vert }^2}& =& {\displaystyle {\Vert \sum _{j=1}^m⟨x,a_j⟩a_j-\sum _{j=1}^n⟨x,a_j⟩a_j\Vert }^2}\\ & =& {\displaystyle {\Vert \sum _{j=m+1}^n⟨x,a_j⟩a_j\Vert }^2}\\ & =& {\displaystyle \sum _{j=m+1}^n{⟨x,a_j⟩}^2}\\ & =& {\displaystyle \mid {\Vert x_n\Vert }^2-{\Vert x_m\Vert }^2\mid }\\ & =& {\displaystyle \mid {\Vert x_n\Vert }^2-y^2+y^2-{\Vert x_m\Vert }^2\mid }\\ & \le & {\displaystyle \mid {\Vert x_n\Vert }^2-y^2\mid +\mid y^2-{\Vert x_m\Vert }^2\mid }\\ & <& {\displaystyle \frac{\epsilon }{2}+\frac{\epsilon }{2}=\epsilon \text{.}}\end{array}$$ So $x_1,x_2,\dots$ is a Cauchy sequence in $H\text{.}$ So $\underset{n\to \infty }{\text{lim}}{x}_n$ exists in $H\text{.}$ So $\sum _{j\in {\mathbb{Z}}_{>0}}⟨x,a_j⟩a_j$ exists in $H\text{.}$ (b) To show: If $x\in H$ then $P\left(x\right)\in \bar{W}\text{.}$ Assume $x\in X\text{.}$ To show: $\sum _{n\in {\mathbb{Z}}_{>0}}⟨x,a_n⟩a_n\in \bar{W}\text{.}$ To show: $\underset{k\to \infty }{\text{lim}}{x}_k\in \bar{W}\text{.}$ To show: $x_k\in W\text{.}$ Since $x_k=\sum _{j=1}^k⟨x,a_j⟩a_j\in \text{span}\{a_1,a_2,\dots \}$ then $x_k\in W\text{.}$ So $P\left(x\right)=\underset{k\to \infty }{\text{lim}}{x}_k\in \bar{W}\text{.}$ (c) To show: If $x\in H$ then $x-P\left(x\right)\in {\bar{W}}^{\perp }\text{.}$ Assume $x\in H\text{.}$ To show: $x-P\left(x\right)\in {\bar{W}}^{\perp }\text{.}$ To show: If $b\in \bar{W}$ then $⟨x-P\left(x\right),b⟩=0\text{.}$ Assume $b\in \bar{W}\text{.}$ Let $b_1,b_2,\dots$ be a sequence in $W$ with $\underset{n\to \infty }{\text{lim}}{b}_n=b\text{.}$ To show: $⟨x-P\left(x\right),b⟩=0\text{.}$ $$\begin{array}{rcl}{\displaystyle ⟨x-P\left(x\right),b⟩}& =& {\displaystyle ⟨x-P\left(x\right),\underset{n\to \infty }{\text{lim}}{b}_n⟩}\\ & =& {\displaystyle \underset{n\to \infty }{\text{lim}}⟨x-P\left(x\right),b_n⟩}\end{array}$$ (since $⟨x-P\left(x\right),·⟩:H\to ℂ$ is continuous) and there exist $\ell \in {\mathbb{Z}}_{>0}$ and $c_1,\dots ,c_{\ell }\in ℂ$ such that $$⟨x-P\left(x\right),b_n⟩=\sum _{k=1}^{\ell }{c}_k⟨x-P\left(x\right),a_k⟩\text{.}$$ Then $$\begin{array}{rcl}{\displaystyle ⟨x-P\left(x\right),a_j⟩}& =& {\displaystyle ⟨x-\underset{k\to \infty }{\text{lim}}{x}_k,a_j⟩}\\ & =& {\displaystyle \underset{k\to \infty }{\text{lim}}⟨x-x_k,a_j⟩\text{(since}\hspace{0.17em}⟨·,a_j⟩:H\to ℂ\hspace{0.17em}\text{is continuous)}}\\ & =& {\displaystyle \underset{k\to \infty }{\text{lim}}(⟨x,a_j⟩-⟨x,a_j⟩)}\\ & =& {\displaystyle 0\text{.}}\end{array}$$ So $$⟨x-P\left(x\right),b⟩=\underset{n\to \infty }{\text{lim}}⟨x-P\left(x\right),b_n⟩=\underset{n\to \infty }{\text{lim}}0=0\text{.}$$ So $x-P\left(x\right)\in W^{\perp }\text{.}$

Notes and References

These are a typed copy of Lecture 36 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on September 26, 2014.

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