Let $\mathcal{L}^2[(0,1)]$ denote the set of $\mathbb{C}$-valued square integrable functions on the interval [0,1]. Prove that $\mathcal{L}^2[(0,1)]$ forms a Hilbert Space.
I believe that I can use the following steps to prove this, but I'm not sure how to write the proofs for them. What do you guys think?
a. Prove that $\mathcal{L}^2[(0,1)]$ is a sub-vector of F([0,1]), the vector space of $\mathbb{C}$-valued functions
b. Let <,>: $\mathcal{L}^2[(0,1)]$ x $\mathcal{L}^2[(0,1)]$ $\rightarrow$ $\mathbb{C}$ denote the inner product defined by
= $\int_{0}^{1} f(x)\overline{g(x)} dx$ - in which i have to show its well-defined, linear, conjugate-symmetric, and positive definite.
c. Show that its a metric space
d. Last, show that every Cauchy sequence in $\mathcal{L}^2[(0,1)]$ converges.
I need help to prove (a)-(d)