Marcus here.
I am currently trying to understand the concept Hilbert space, specifically for squared integrable functions. The definition of a hilbert space is "A inner product space $( \langle \cdot , \cdot \rangle, E)$ is said to be complete if every cauchy sequence is convergent. Then $( \langle \cdot , \cdot \rangle, E)$ is also said to be a hilbert space". A standard example is that the set of squared integrable functions is a hilbert space, that is \begin{align*} L^2(\Omega) = \{ f:\Omega \to \mathbb{C}:\int_{\Omega} |f|^2 dx < \infty\} \end{align*} For $f,g \in L^2(\Omega)$ we define the inner product $\langle f , g \rangle = \int_{\Omega}f^*g \ dx$, where $f^*$ is the complex conjugated of $f$, this inner product space, $( \langle \cdot , \cdot \rangle, L^2(\Omega))$, forms a hilbert space. I try to use the hilbert space theory to understand the wave equation equipped with the dirichlet boundary condition on $\Omega$ for a drum, thus in my case I am interested in the case where $\Omega \subseteq \mathbb{R}^2$. But I am stuck with the basic question why is $( \langle \cdot , \cdot \rangle, L^2(\Omega))$ a hilbert space in the first place? I do not know much about functional analysis and measure theory, thus I am interested in a proof using basic principles, so the questions is do such a basic proof exist?