MATH 323 Lecture 22

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Inner Product and Norm

Cauchy-Schwarz Inequality

$|\langle u,v\rangle |\leq \|u\|\|v\|$

Can be rewritten as

-1\leq \underbrace {\frac {\langle u,v\rangle }{\|u\|\|v\|}} _{\cos {\theta }}\leq 1

$V$ is normed linear space if with each $v\in V$ , the number \|v\| is associated:

$\|v\|\geq 0$ and $\|v\|=0\iff v={\vec {0}}$
$\|\alpha v\|=|\alpha |\|v\|\quad \forall \alpha \in \mathbb {R}$
$\|v+w\|\leq \|v\|+\|w\|$ (triangle inequality)

Theorem 5.4.3

If $V$ is an inner product space, then $\|v\|={\sqrt {\langle v,v\rangle }}$ is a norm.

Euclidean Norm

$\mathbb {R} ^{n}$ , $\langle x,y\rangle =x\cdot y$ , $\|x\|={\sqrt {\langle x,x\rangle }}$ is a norm

Euclidean Norm given by $\|x\|_{p}$ is defined for $p\geq 1$ as

\|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{\frac {1}{p}}

Supreme norm $\|x\|_{\infty }=\max _{1\leq i\leq n}|x_{i}|$

Example

For ${\vec {x}}=\langle 4,-5,3\rangle \in \mathbb {R} ^{3}$

$\|x\|_{1}=4+5+3=12$
$\|x\|_{2}={\sqrt {16+25+9}}=5{\sqrt {2}}$
$\|x\|_{\infty }=5$

Applications in Orthogonality

Inner product space $V$ , $\langle \cdot ,\cdot \langle$

For vectors $\{v_{1},\ldots ,v_{n}\}\subset V$ such that $\langle v_{i},v_{j}\rangle =0$ when $i\neq j$ , $\{v_{1},\ldots ,v_{n}\}$ is an orthogonal set of vectors.

If we transform $\{v_{1},\ldots ,v_{n}\}$ into a set of unit vectors $\{u_{1},\ldots ,u_{n}\}$ , then

\langle u_{i},u_{j}\rangle =\delta _{ij}={\begin{cases}0&i\neq j\\1i=j\end{cases}}

This is called an orthonormal set

Theorem 5.5.1

If $\{v_{1},\ldots ,v_{n}\}$ is an orthogonal set of nonzero vectors, then the vectors are linearly independent.

${\begin{aligned}\langle v_{j},c_{1}\,v_{1}+\dots +c_{n}\,v_{n}\rangle &=\langle v_{j},0\rangle =0\\&=c_{1}\langle v_{j},v_{1}\rangle +\dots +c_{j}\langle v_{j},v_{j}\rangle +\dots +c_{n}\langle v_{j},v_{n}\rangle \\&=c_{j}\langle v_{j},v_{j}\rangle =0\iff c_{j}=0\end{aligned}}$

Theorem 5.5.2

An orthonormal set $B=\{u_{1},\ldots ,u_{k}\}$ is a basis for $S=\mathrm {Span} \{u_{1},\ldots ,u_{k}\}$ and is called an orthonormal basis

Let $\{u_{1},\ldots ,u_{n}\}$ be an orthonormal basis for $V$ .

V=\sum _{i=1}^{n}c_{i}\,u_{i}\quad \implies \quad c_{i}=\langle v,u_{i}\rangle

Corollary

For an orthonormal basis $\{u_{1},\ldots ,u_{n}\}$ ,

v=\sum _{i=1}^{n}a_{i}\,u_{i}\quad {\text{and}}\quad w=\sum _{i=1}^{n}b_{i}\,u_{i}\quad \implies \quad \langle v,w\rangle =\sum _{i=1}^{n}a_{i}\,b_{i}

Corollary: Parseval's Formula