A Hilbert space \(\mathcal{H}\) is an inner product space that is separable and -complete. Separable means and complete means that the limits of all Cauchy Sequences are also in \(\mathcal{H}\) e.g. \(\mathbb{R}\) is ok, but \(\mathbb{Q}\) is not.
The finite-dimensional reals \(\mathbb{R}^D\) with inner product \(\langle u, v \rangle := \sum_{d=1}^D u_d v_d^*\)
\(\ell^2 := \{ u \in \mathbb{R}^{\mathbb{N}} : \sum_n u_n^2 < \infty\) with inner product \(\langle u, v \rangle := \sum_{n} u_n v_n^*\)
\(L^2([0, T]) := \{ f: X \rightarrow \mathbb{C} : \int_0^T |f(x)|^2 dx < \infty\) with inner product \(\langle u, v \rangle := \int u(x) v(x)^* dx\)
\(L^2\) is the only Hilbert space of the \(L^p\) spaces.
Any Hilbert space possess an orthonormal basis \(\{ \phi_n \}_{n \geq 1}\) which satisfies \(\langle \phi_n, \phi_m \rangle = \delta_{n, m}\), where \(\delta_{n, m}\) is the Kronecker delta.
A consequence of Hilbert spaces possessing orthonormal bases is that any function in a space can be expressed as a linear combination of that space’s basis.
All elements of a Hilbert space can create linear functionals in that space. Specifically, if \(f \in \mathcal{H}\), then we can define a functional \(F: \mathcal{H} \rightarrow \mathbb{C}\) as \(F_f(g) := \langle f, g \rangle\)
Relatedly, for any continuous linear functional \(F: \mathcal{H} \rightarrow \mathbb{C}\), there exists an element \(f\) in the Hilbert space such that the functional is given by the inner product: \(F(g) = \langle f , g \rangle\)
Continuity is necessary! Otherwise, this doesn’t work. For example, suppose we want to find \(f\) to construct the functional \(F_f(g) := g(x)\). We know that we must be able to write this functional as an inner product \(\int g(t) f(t)^* dt\), but the only function that can result in \(g(x) = \int g(t) f(t)^* dt\) is the Dirac delta function \(\delta(x)\), but it doesn’t belong to the Hilbert space!
One common orthonormal basis for the Hilbert space \(L^2([0, T])\) is the Fourier basis, defined as:
\[\phi_z(t) := \frac{1}{\sqrt{T}} e^{i \frac{2 \pi z}{T} t}\]This basis is orthonormal because
\[\begin{align*} \langle \phi_n, \phi_m \rangle_{\mathcal{H}} &= \frac{1}{T} \int_{0}^T e^{i \frac{2 \pi n}{T} t} e^{-i \frac{2 \pi m}{T} t} dt\\ &= \begin{cases}1 & n = m \\ 0 & n \neq m \end{cases} \end{align*}\]Once you fix a Hilbert’s space’s basis, any element in the Hilbert space can be equivalently thought of as a sequence of coefficients of the element projected onto each element of the basis:
\[f := \sum_{z=-\infty}^{\infty} \langle f, \phi_z \rangle_H \phi_z\]with the inner product redefined as:
\[\langle f, g \rangle_H = \sum_z \langle f, \phi_z \rangle_H \langle g, \phi_z \rangle_H\]The implication of this is that once you fix a Hilbert space’s basis, \(H\) becomes equivalent to \(\ell_2\) because all functions/vectors look the same: sequences of basis coefficients.