Hilbert Spaces
See Sections 1-5 of these lecture notes for a gentle introduction to the notion of Hilbert spaces.
1. Preliminaries: Inner Product Spaces
Definition of an Inner Product Suppose \(V\) is a vector space over a field \(F\) which is either real or complex. We say that a pairing \(\left< \cdot , \cdot \right> \ : V \times V \rightarrow F\) is an inner product when it satisfies the following criteria:
- (Sesquilinearity) For any \(f,g,h \in V\) and \(c \in F\),
- \(\left< c f + g, h\right> = c \left<f,h\right>+ \left<g,h\right>\)
- \(\left<f, h\right> = \left< h , f \right>\) when the field \(F\) is real and \(\left< f, h \right> = \overline{\left< h , f \right>}\) when \(F\) is complex.
- (Nonnegativity) For any \(f \in V\), \(\left<f,f\right>\) is real and nonnegative. Moreover, \(\left<f,f\right> = 0\) if and only if \(f = 0\).
A vector field \(V\) with an inner product \(\left<\cdot,\cdot\right>\) are together referred to as an inner product space.
Theorem (Cauchy-Schwarz Inequality)
Any inner product \(\left< \cdot , \cdot \right>\) satisfies
\[ \left| \left<f , g \right> \right| \leq \sqrt{ \left< f,f\right> \left<g,g\right> }\]
with equality if and only if \(f\) and \(g\) are linearly dependent.
Proof
Corollary
If \(||f|| := \sqrt{\left<f,f\right>}\), then \(||\cdot||\) is a norm on \(V\). In particular, it satisfies the triangle inequality:
\[ || f + g || \leq ||f|| + ||g|| \]
for all \(f,g \in V\), with equality if and only if \(f\) and \(g\) point in the same direction (i.e., if there exists \(h\) such that \(f\) and \(g\) are both nonnegative scalar multiples of \(h\).)
Proof
2. Hilbert Spaces: Definition and Examples
Definition of a Hilbert Space A Hilbert space \(H\) is an inner product space \((H,\left<\cdot,\cdot\right>)\) which is complete with respect to the associated metric \(d(f,g) := ||f-g||\).
Example Hilbert Space: \({\mathbb R}^n\) Suppose that vectors \(x\) and \(y\) in \({\mathbb R}^n\) have standard coordinates \((x_1,\ldots,x_n)\) and \((y_1,\ldots,y_n)\), respectively. Let \(\left<x,y\right> := x_1 y_1 + \cdots + x_n y_n\). This \(\left<\cdot,\cdot\right>\) defines an inner product on \({\mathbb R}^n\); the topology induced by the norm is the standard topology, and in particular \({\mathbb R}^n\) is a Hilbert space when equipped with this inner product.
Example: Incompleteness Let \(V\) be the vector space of continuous, complex-valued functions \(f\) on the unit interval \([-1,1]\). The functional
\[ \left<f,g\right> := \int_{-1}^1 f(t) \overline{g(t)} dt \]
defines an inner product on \(V\), but \(V\) is not complete and consequently \((V,\left<\cdot,\cdot\right>)\) is an inner product space but not a Hilbert space.
Exercises
- Suppose that \(\{x_n\}_{n=1}^\infty\) and \(\{y_n\}_{n=1}^\infty\) are convergent sequences in a Hilbert space \(H\). Show that \(\lim_{n \rightarrow \infty} \left<x_n,y_n\right>\) exists and\[ \lim_{n \rightarrow \infty} \left<x_n,y_n\right> = \left<\lim_{n \rightarrow \infty} x_n, \lim_{n \rightarrow \infty} y_n\right>. \]HintBegin by rewriting the difference \(\left<x_n,y_n\right>-\left<x,y\right>\) to more prominently feature the differences \(x_n - x\) and \(y_n - y\).
- We say that a sequence \(\{x_n\}_{n=1}^\infty\) in a Hilbert space converges weakly to \(x \in H\) when \(\lim_{n \rightarrow \infty} \left<x_n,y\right> = \left<x,y\right>\) for each \(y \in H\). Prove that \(x_n \rightarrow x\) in the standard topology of \(H\) if and only if \(\{x_n\}_{n=1}^\infty\) converges weakly to \(x\) and \(||x_n|| \rightarrow ||x||\) as \(n \rightarrow \infty\).HintFor the reverse direction, expand \(||x_n - x||^2\) and use the facts that \(\left<x_n,x\right> \rightarrow ||x||^2\) and \(||x_n||^2 \rightarrow ||x||^2\) as \(n \rightarrow \infty\) to show that \(||x_n - x||^2 \rightarrow 0\) as \(n \rightarrow \infty\).