Definition 5.4.1. Inner Product Space.
A vector space \(V\) with an inner product is called an inner product space.
Section 5.4 Inner product space
Subsection 5.4.1 Abstract Vector Space
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Let \(\mathbf{P}_n(x)=\{a_0+a_1x+\ldots+a_nx^n|a_i\in \mathbb{R}\}\) is a vector space with two operations\begin{align*} (a_0+a_1x \amp +\ldots+a_nx^n) +(b_0+b_1x+\ldots+b_nx^n) \\ \amp = (a_0+b_0)+(a_1+b_1)x+\ldots+(a_n+b_n)x^n \end{align*}\begin{equation*} k(a_0+a_1x+\ldots+a_nx^n)=(k a_0)+(k a_1)x+\ldots+(k a_n)x^n \end{equation*}
- Define \(C[a, b]\) be the set of all real-valued continuous functions defined on the interval \([a,b]\text{.}\) \(C[a,b]\) is a vector space with operations\begin{equation*} (f+g)(x)=f(x)+g(x) \text{ and } (c f)(x)=c[f(x)]\text{.} \end{equation*}
Subsection 5.4.2 Inner Product Space
Let \(\mathbf{u}, \mathbf{v}\text{,}\) and \(\mathbf{w}\) be vectors in a vector space \(V\text{,}\) and let \(c\) be any scalar. An inner product on \(V\) is a function that associates a real number \(\langle\mathbf{u}, \mathbf{v}\rangle\) with each pair of vectors \(\mathbf{u}\) and \(\mathbf{v}\) and satisfies the axioms listed below.
- \(\displaystyle \langle\mathbf{u}, \mathbf{v}\rangle=\langle\mathbf{v}, \mathbf{u}\rangle\)
- \(\displaystyle \langle\mathbf{u}, \mathbf{v}+\mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{v}\rangle+\langle\mathbf{u}, \mathbf{w}\rangle\)
- \(\displaystyle c\langle\mathbf{u}, \mathbf{v}\rangle=\langle c \mathbf{u}, \mathbf{v}\rangle\)
- \(\langle\mathbf{v}, \mathbf{v}\rangle \geq 0\text{,}\) and \(\langle\mathbf{v}, \mathbf{v}\rangle=0\) if and only if \(\mathbf{v}=\mathbf{0}\text{.}\)
Lemma 5.4.2.
- In \(\mathbf{P}_n(x)\text{,}\) the function\begin{equation*} \langle p_1(x), p_2(x) \rangle=a_0b_0+a_1b_1+\ldots+a_nb_n \end{equation*}is an inner product of \(\mathbf{P}_n(x)\text{,}\) where \(p_1(x)=a_0+a_1x+\ldots+a_nx^{n}, p_2(x)=b_0+b_1x+\ldots+b_nx^{n}.\)
- In \(C[a,b]\text{,}\) the function\begin{equation*} \langle f(x),g(x)\rangle=\int\limits_{a}^{b} f(x)g(x)dx \end{equation*}is an inner product of \(C[a,b]\text{.}\)
Subsection 5.4.3 Gram-Schmidt Process in an Inner Product Space
Proposition 5.4.3.
- \(\mathbf{P}_n(x)\) with the inner product in Lemma 5.4.2 is an inner product space.
- \(C[a,b]\) with the inner product in Lemma 5.4.2 is an inner product space.
Theorem 5.4.4.
Let \(S=\{\mathbf{v}_1,\mathbf{v}_2,\ldots, \mathbf{v}_n\}\) be a basis of an inner product space \(V\text{.}\)
- Let \(B^{\prime}=\left\{\mathbf{w}_{1}, \mathbf{w}_{2}, \ldots, \mathbf{w}_{n}\right\}\text{,}\) where\begin{align*} w_1\amp=\mathbf{v}_{1} \\ w_2\amp=\mathbf{v}_{2}-\frac{\langle\mathbf{v}_{2}, \mathbf{w}_{1}\rangle}{\langle \mathbf{w}_{1}, \mathbf{w}_{1}\rangle} \mathbf{w}_{1}\\ w_3\amp=\mathbf{v}_{3}-\frac{\langle\mathbf{v}_{3}, \mathbf{w}_{1}\rangle}{\langle \mathbf{w}_{1}, \mathbf{w}_{1}\rangle} \mathbf{w}_{1}-\frac{\langle\mathbf{v}_{3}, \mathbf{w}_{2}\rangle}{\langle \mathbf{w}_{2}, \mathbf{w}_{2}\rangle} \mathbf{w}_{2} \\ \amp\vdots\\ w_m\amp=\mathbf{v}_{n}-\frac{\langle\mathbf{v}_{n}, \mathbf{w}_{1}\rangle}{\langle \mathbf{w}_{1}, \mathbf{w}_{1}\rangle} \mathbf{w}_{1}-\frac{\langle\mathbf{v}_{n}, \mathbf{w}_{2}\rangle}{\langle \mathbf{w}_{2}, \mathbf{w}_{2}\rangle} \mathbf{w}_{2}-\ldots-\frac{\langle\mathbf{v}_{n}, \mathbf{w}_{n-1}\rangle}{\langle \mathbf{w}_{n-1}, \mathbf{w}_{n-1}\rangle} \mathbf{w}_{n-1} \end{align*}\(B'\) is an orthogonal basis of \(W\text{.}\)
- Let \(\mathbf{u}_{i}=\frac{\mathbf{w}_{i}}{\left\|\mathbf{w}_{i}\right\|}\text{.}\) Then \(B^{\prime \prime}=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{n}\right\}\) is an orthonormal basis for \(W\text{.}\) Also, for \(k=1,2, \ldots, m\text{,}\)\begin{equation*} \operatorname{span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\}=\operatorname{span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{k}\right\} \end{equation*}
Example: In \(C[0,2]\text{,}\) find an orthonormal basis of the space \(W=\operatorname{span}\{1,x,x^{2}\}\text{.}\)
