17  Hilbert spaces

17.1 \(L^2\) – the square integrable functions

For a quantum chemist, the most important \(L^p\) space is \(L^2(\Omega)\), where \(\Omega\) is a measure space, such as \(\mathbb{R}^3\), or \(X = \mathbb{R}^3 \times \{\uparrow,\downarrow\}\). Previously, we saw that this was a Banach space with the \(L^2\) norm, but this norm comes from an inner product: \[\left\langle f,g\right\rangle_{L^2} := \int_\Omega \overline{f(x)} g(x) \, \mathrm{d}x,\] where the complex conjugate is relevant only if \(\mathbb{F}= \mathbb{C}\). Again, functions that agree on sets of measure zero are identified.

17.2 \(\ell_2\) – the archetypal separable Hilbert space

Consider the following situation: Let \(u = (u_i) \subset \mathbb{F}\) be a real or complex-valued sequence. Let \(p \in [1,+\infty)\). We can define a norm given by \[\|u\|_p := \left(\sum_{i\in\mathbb{N}} |u_i|^p \right)^{1/p}.\] The set of sequences such that \(\|u_p\|\) is a convergent sum is called \(\ell_p\). For \(p=+\infty\), we set \(\|u\|_{+\infty} = \max_i |u_i|\). The spaces \(\ell_p\) are all Banach spaces.

In particular, the space \(\ell_2\) is a Hilbert space when we use the inner product \[\left\langle u,v\right\rangle = \sum_{i\in\mathbb{N}} \overline{u_i} v_i.\] This Hilbert space is an archetypal Hilbert space, as we will next see.

17.3 Existence of orthonormal bases

The following fact is significant, because it tells us that \(\ell_2(\mathbb{N};\mathbb{F})\) is an archetypal separable Hilbert space, much in the same manner as \(\mathbb{F}^n\) is the archetypal finite-dimensional Hilbert space.

Definition 17.1 (Orthnormal basis for separable Hilbert space) An orthonormal basis for an infinite dimensional separable Hilbert space \(V\) is a linearly independent orthonormal set \(\{b_i\}\subset V\) (i.e., \(\left\langle b_i,b_j\right\rangle = 0\) whenever \(i\neq j\), and \(\left\langle b_i,b_i\right\rangle = 1\)), such that for every \(u \in V\), there exist numbers \(c_i \in \mathbb{F}\) such that \[u = \sum_{i=1}^\infty c_i b_i.\] The infinite sum is to be interpreted as a series, i.e., \(u \in V\) means \[\|u - \sum_{i=1}^N c_i b_i\| \to 0 \quad \text{as} \quad N\to+\infty.\]

The space of sequences equipped with the Euclidean inner product is denoted \(\ell_2(\mathbb{N})\), \[\left\langle c,d\right\rangle_{\ell_2} = \sum_{i=1}^\infty \overline{c_i} d_i.\] We see that \(V\) and \(\ell_2\) are isometrically isomorphic when a basis is given, since if \(v = \sum_i d_i b_i\) then \(\left\langle u,v\right\rangle= \left\langle c,d\right\rangle_{\ell_2}\).

An fundamental result is the following:

Theorem 17.1 (Statement) Any separable Hilbert space has an orthonormal basis.

Thus any separable Hilbert space is essentially \(\ell_2(\mathbb{N})\) after a basis has been chosen.