A Hilbert space, named after the German mathematician David Hilbert, is a complete inner product space. It’s a vector space with an inner product that allows you to define angles and distances between vectors, and it is complete in the sense that all Cauchy sequences in the space converge to a point in the space. Mathematically, a Hilbert space ℋ is a vector space over the field ℝ (real numbers) or ℂ (complex numbers) with an inner product (⋅, ⋅ ) and is complete with respect to the norm induced by the inner product.
In more precise terms, a Hilbert space is a set of elements such that:
1. It is a vector space: For any two elements u, v ∈ ℋ and scalars α, β ∈ ℝ or ℂ, it holds that αu + βv ∈ ℋ.
2. It has an inner product: An inner product is a function (⋅, ⋅ ) : ℋ × ℋ → ℝ or (⋅, ⋅ ) : ℋ × ℋ → ℂ that satisfies the following properties:
a. Linearity: (αu + βv, w) = α(u, w) + β(v, w) for any u, v, w ∈ ℋ and α, β ∈ ℝ or ℂ.
b. Conjugate symmetry: $(u, v) = \overline{(v, u)}$ for any u, v ∈ ℋ, where $\overline{\cdot}$ denotes the complex conjugate.
c. Positive-definite: (u, u) ≥ 0 for any u ∈ ℋ, with equality if and only if u = 0.
3. It is complete: A space is complete if every Cauchy sequence of elements in the space converges to a limit in the space. In other words, for any sequence of elements un in ℋ, if ∀ϵ > 0, ∃N such that ∥um − un∥ < ϵ for all m, n > N, then there exists an element u ∈ ℋ such that un → u as n → ∞.
One of the simplest examples of a Hilbert space is ℝn, the n-dimensional Euclidean space with the usual dot product as the inner product:
$$(u, v) = u \cdot v = \sum_{i=1}^n u_i v_i$$
This space is complete, and the induced norm is the standard Euclidean norm:
$$\|u\| = \sqrt{(u, u)} = \sqrt{\sum_{i=1}^n u_i^2}$$
In the context of mathematical finance, Hilbert spaces are often used to describe the space of square-integrable functions, which is essential for the study of stochastic processes, such as Brownian motion and other continuous-time models. A commonly used Hilbert space in finance is the L2 space of square-integrable functions:
L2([a, b]) = {f:[a,b]→ℝor ℂ | ∫ab|f(x)|2dx<∞}
The inner product in this space is defined as:
$$(f, g) = \int_a^b f(x) \overline{g(x)} dx$$
This space is complete under the induced norm:
$$\|f\| = \sqrt{(f, f)} = \sqrt{\int_a^b |f(x)|^2 dx}$$