The Hilbert space ${\cal H}= {\cal H}_{\rm well}$

This is the space of of square integrable wave functions $\psi(x)$ of the infinite potential well, section 2.2, defined on the interval $[0,L]$, with $\psi(x)=\psi(L)=0$. Each function $\psi$ is considered as a vector, the linear structure of a vector space comes from the fact the $\psi \in {\cal H},\phi \in {\cal H} \leadsto \psi + \phi, c\psi \in {\cal H}$. The scalar product is given by an integral

$$\langle \psi \vert \phi \rangle := \int_0^Ldx \psi^*(x)\phi(x).$$ (170)

You can check that this is a scalar product indeed. The eigenvectors of the Hamiltonian $\hat{H}$, i.e. the functions $\phi_n(x)$ with energy $E_n$,

$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right ),\quad E_n= \frac{n^2 \hbar^2 \pi^2}{2mL^2},\quad n=1,2,3,...$$ (171)

form an orthonormal basis of ${\cal H}$

$$\langle \psi_n \vert \psi_m \rangle =\delta_{nm}.$$ (172)

Any wave function $f \in {\cal H}$ can be expanded into a linear combination of basis vectors, i.e. eigenfunctions $\psi_n$,

$$f=\sum_{n=1}^{\infty}\langle \psi_n \vert f \rangle \psi_n.$$ (173)

We summarize the above two examples in the following table:

example	$R^d$	${\cal H}_{\rm well}$
vector	x	wave function $\psi$
scalar product	$\langle {\bf x} \vert {\bf y} \rangle = \sum_{n=1}^d x_n^*y_n$	$\langle \psi \vert \phi \rangle := \int_0^Ldx \psi^*(x)\phi(x)$
orthonormal basis	$\langle {\bf e}_n \vert {\bf e}_m \rangle=\delta_{nm}$	$\langle \psi_n \vert \psi_m \rangle =\delta_{nm}$
completeness	${\bf x}=\sum_{n=1}^d\langle {\bf e}_n\vert{\bf x} \rangle{\bf e}_n$	$\psi =\sum_{n=1}^{\infty}\langle \psi_n \vert \psi \rangle \psi_n$
components	${\bf x} = (\langle {\bf e}_1\vert{\bf x} \rangle,...,\langle {\bf e}_d\vert{\bf x} \rangle)$	$\psi = (\langle \psi_1 \vert \psi \rangle, \langle \psi_2 \vert \psi \rangle,...)$

If you understand this table, you are already halfway in completely understanding the math underlying quantum mechanics.