91Ó°ÊÓ

Let \(C^{1}[0,1]\) be the space of continuous functions \(f:[0,1] \rightarrow \mathbb{C}\) with a continuous first derivative on \([0,1]\). (a) Prove that $$ \langle f, g\rangle=f(0) \cdot \overline{g(0)}+\int_{0}^{1} f^{\prime}(x) \overline{g^{\prime}(x)} d x $$ is an inner product on \(C^{1}[0,1]\). (b) Find an orthonormal system \(\left\\{h_{1}, h_{2}, h_{3}\right\\}\) in \(C^{1}[0,1]\), with respect to this inner product, for which $$ \operatorname{span}\left\\{h_{1}, h_{2}, h_{3}\right\\}=\operatorname{span}\left\\{1, x, x^{2}\right\\} . $$

The Legendre polynomials are defined by $$ P_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n} $$ (a) Prove that $$ \int_{-1}^{1} P_{m}(x) P_{n}(x) d x= \begin{cases}0, & m \neq n \\ \frac{2}{2 n+1}, & m=n\end{cases} $$ and therefore \(\left\\{P_{n}\right\\}_{n=1}^{\infty}\) is an infinite orthogonal system on the space \(C[-1,1]\) with inner product $$ \langle f, g\rangle=\int_{-1}^{1} f(x) \overline{g(x)} d x . $$ (b) Let $$ f(x)= \begin{cases}0, & -1 \leq x \leq 0 \\ 1, & 0

Let \(V\) be an inner product space. Prove that for all \(u, v \in V\) the "Parallelogram law" $$ \|u+v\|^{2}+\|u-v\|^{2}=2\|u\|^{2}+2\|v\|^{2} $$ holds.