Spectral decomposition of $-\Delta$ https://ift.tt/eA8V8J

I am currently trying to self learn about an interesting idea that caught my eye in spectral geometry, which is the whole idea of hearing the shape of the drum by solving the wave equation $\Delta \psi = k^2 \psi_{tt}$ on a compact Riemannian manifold $(M,g)$ with dirichlet boundary constraint $\psi|_{\partial M}=0$. The thing is, I don't think I have that of a solid background in functional analysis and PDEs, and I am looking for some help or a rather self contained source which proves that such decomposition exists $$ -\Delta(\cdot)= \sum_{l=0}^\infty \lambda_l \left \langle \cdot,\phi_l \right \rangle_{L^2(M)}\phi_l$$ What I do know: I wans't able to find many sources which dive deep into this problem, but the first thing I was able to realize is this decomposition probably isn't on $L^2(M)$ but rather some subspace of the form $V=\left \{ f\ \text{nice enough}|\ f|_{\partial M}=0\right \}$, the reason for this is that we probably need the laplacian to be self adjoint, and looking at one of Green's identities $$\int_M u\Delta v - v\Delta u =\int_{\partial M}u\frac{\partial v}{\partial n}-v \frac{\partial u}{\partial n}$$ We need to zero the RHS with the constraint. What is $V$? I saw alot of references to a "Sobolev space $H_0^1(M)$" but couldn't find a coherent definition of it, in addition to the fact that it is equipped with a different inner product than of $L^2(M)$ which makes things more confusing for me.

Anyhow, this tells us automatically why eigenfunctions must be orthogonal, since if we have $$ -\Delta \phi_1 = \lambda_1 \phi_1,\ -\Delta \phi_2 = \lambda_2 \phi_2$$ for $\lambda_1,\lambda_2 \neq 0$ different then $$\left \langle \lambda_1\phi_1,\phi_2 \right \rangle_{L^2(M)}=-\left \langle \Delta\phi_1,\phi_2 \right \rangle_{L^2(M)}=-\left \langle \phi_1,\Delta\phi_2 \right \rangle_{L^2(M)}=\left \langle \phi_1,\lambda_2\phi_2 \right \rangle_{L^2(M)}$$ and we get that $\left \langle \phi_1,\phi_2 \right \rangle_{L^2(M)}=0$

I also know that $-\Delta$ is positive definite since $$\int_M u (-\Delta) u=\int_M \left \| \nabla u \right \|_g^2\geq 0$$

What I'm looking for: This topic is pretty weird for me right now and I would like to learn about it thoroughly.

1. Are my proofs correct or relevant at all?
2. As I mentioned what is $V$? it's hard for me to see why $C^2(M) \cap C(\bar{M})$ needs to be restricted further.
3. How do we know that $-\Delta$ is compact if at all? This means that if $f_k \rightarrow f$ in $V$, then $-\Delta f_k$ has a convergent subsequence of functions. I know according to functionaly analysis this should imply the spectral decomposition but I don't know whether this is true.
4. Assuming 3 is true, we also get for free that the eigenvalues are discrete $0\leq \lambda_0 \leq \lambda_1 \leq \cdots$. However, I saw claims that $\lambda_k \rightarrow \infty$ (when according to functional analysis it should approach to $0$). This doesn't make sense to met, how sums like the spectral decomposition $\Delta f=\sum_{l=0}^\infty \lambda_l\left \langle f,\phi_l \right \rangle_{L^2(M)}\phi_l $ converge?
5. How does one show $\left \{ \phi_l\right \}_{l=0}^\infty$ is complete? meaning we can express any $f\in V$ as a fourier series $f=\sum_{l=0}^\infty \left \langle f,\phi_l \right \rangle_{L^2(M)}\phi_l $

Sorry for the mess of a question, In the probable case I said something wrong I would love to hear. If anyone knows of a free-access source which adresses this problem please refer me to it!

from Hot Weekly Questions - Mathematics Stack Exchange
Theorem