MATH 323 Lecture 22

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Inner Product and Norm

Cauchy-Schwarz Inequality

$|⟨u,v⟩|\le \Vert u\Vert \Vert v\Vert$

Can be rewritten as

$-1\le \underset{\cos \theta }{\underbrace{\frac{⟨u,v⟩}{\Vert u\Vert \Vert v\Vert }}}\le 1$

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

1. $\Vert v\Vert \ge 0$ and $\Vert v\Vert =0⟺v=\overrightarrow{0}$
2. $\Vert \alpha v\Vert =|\alpha |\Vert v\Vert \mathrm{\forall }\alpha \in \mathbb{ℝ}$
3. $\Vert v+w\Vert \le \Vert v\Vert +\Vert w\Vert$ (triangle inequality)

Theorem 5.4.3

If $V$ is an inner product space, then $\Vert v\Vert =\sqrt{⟨v,v⟩}$ is a norm.

Euclidean Norm

${\mathbb{ℝ}}^n$, $⟨x,y⟩=x\cdot y$, $\Vert x\Vert =\sqrt{⟨x,x⟩}$ is a norm

Euclidean Norm given by $\Vert x{\Vert }_p$ is defined for $p\ge 1$ as

$\Vert x{\Vert }_p={(\sum _{i=1}^n|x_i{|}^p)}^{\frac{1}{p}}$

Supreme norm $\Vert x{\Vert }_{\mathrm{\infty }}=\underset{1\le i\le n}{\max }|x_i|$

Example

For $\overrightarrow{x}=⟨4,-5,3⟩\in {\mathbb{ℝ}}^3$

• $\Vert x{\Vert }_1=4+5+3=12$
• $\Vert x{\Vert }_2=\sqrt{16+25+9}=5\sqrt{2}$
• $\Vert x{\Vert }_{\mathrm{\infty }}=5$

Applications in Orthogonality

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

For vectors $\{v_1,\dots ,v_n\}\subset V$ such that $⟨v_i,v_j⟩=0$ when $i\ne j$, $\{v_1,\dots ,v_n\}$ is an orthogonal set of vectors.

If we transform $\{v_1,\dots ,v_n\}$ into a set of unit vectors $\{u_1,\dots ,u_n\}$, then

$⟨u_i,u_j⟩={\delta }_{ij}=\{\begin{array}{ll}0& i\ne j\\ 1i=j\end{array}$

This is called an orthonormal set

Theorem 5.5.1

If $\{v_1,\dots ,v_n\}$ is an orthogonal set of nonzero vectors, then the vectors are linearly independent.

$\begin{array}{rl}⟨v_j,c_1{v}_1+\dots +c_n{v}_n⟩& =⟨v_j,0⟩=0\\ & =c_1⟨v_j,v_1⟩+\dots +c_j⟨v_j,v_j⟩+\dots +c_n⟨v_j,v_n⟩\\ & =c_j⟨v_j,v_j⟩=0⟺c_j=0\end{array}$

Theorem 5.5.2

An orthonormal set $B=\{u_1,\dots ,u_k\}$ is a basis for $S=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}\{u_1,\dots ,u_k\}$ and is called an orthonormal basis

Let $\{u_1,\dots ,u_n\}$ be an orthonormal basis for $V$.

$V=\sum _{i=1}^n{c}_i{u}_i⟹c_i=⟨v,u_i⟩$

Corollary

For an orthonormal basis $\{u_1,\dots ,u_n\}$,

$v=\sum _{i=1}^n{a}_i{u}_i\text{and}w=\sum _{i=1}^n{b}_i{u}_i⟹⟨v,w⟩=\sum _{i=1}^n{a}_i{b}_i$

Corollary: Parseval's Formula

For an orthonormal basis $\{u_1,\dots ,u_n\}$,

$v=\sum _{i=1}6n{c}_i{u}_i⟹\Vert v{\Vert }^2=\sum _{i=1}^n{c}_i^2$