So, I would appreciate any critiques I can get for my proof of the following problem.
The Problem:
Let $[a.b]$ be the closed, bounded interval of real numbers. Show that the $L^2[a,b]$ inner product is also an inner product on $C[a,b]$. Is $C[a,b]$ considered as an inner product space with the $L^2[a,b]$ inner product, a Hilbert space?
Proof: To prove that the inner product of the $L^2[a,b]$ space is also the inner product of the $C[a,b]$ space of continuous functions on $[a,b]$, We check that the three properties of the inner product hold on $C[a,b]$.
(i) \begin{align*} \langle \alpha x_1+\beta x_2, y\rangle &=\int^b_a \left(\alpha x_1+\beta x_2\right)y=\int^b_a\alpha x_1\cdot y+\int^b_a\beta x_2\cdot y\\ &=\alpha\int^b_a x_1\cdot y+\beta\int^b_a x_2\cdot y=\alpha\langle x_1,y\rangle+\beta\langle x_2,y\rangle. \end{align*} (ii) \begin{align*} \langle x,y\rangle=\int^b_a xy=\int^b_a yx=\langle y,x\rangle. \end{align*} (iii) \begin{align*} \langle x,x\rangle=\int^b_a x^2\geq 0\quad \text{with equality occuring at}\quad x=0. \end{align*} Since the three properties hold, $C[a,b]$ is indeed an inner product space with the inner product defined on $L^2$.
We must now show that the space $C[a,b]$ is not complete with respect to the norm induced by the inner product which is $\|h\|=\sqrt{\langle h,h\rangle}$. We shall write this out as, $$ \|h\|^2=\left(\int^b_a |h|^2\right). $$ We need to find some Cauchy sequence of functions $\{f_n\}$ such that it does not converge in $C[a,b]$. Consider the sequence of functions defined as, $$ f_n (x) :=\begin{cases} 1, & \text{if}\;x\geq t,\\ 0, & \text{if}\;x\leq t-\frac{1}{n},\\ nx-tn+1, & \text{if}\;t-\frac{1}{n}<x<t, \end{cases} $$ where $t\in (a,b)$ is fixed. We know such a function is a continuous function on $[a,b]$. WLOG suppose that $n>m$, we then have $$ \|f_n-f_m\|_{L_2}^2=\int_{t-\frac{1}{m}}^t |f_n-f_m|^2\,dx\leq \frac{1}{m}<\frac{1}{N}. $$ We select $N=\left\lceil\frac{1}{\epsilon^2}\right\rceil$ and we see that this is indeed a Cauchy Sequence. However, $\lim_{n\rightarrow\infty}f_n(x)$ is actually a discontinuous function. So, this is an example where convergence of a Cauchy sequence in $C[a,b]$ does not converge in $C[a,b]$ and thus, it is not a complete space. Therefore, it is not a Hilbert space.