140 Hilbert Space
A set of vectors with an inner product that induces a norm and in which every Cauchy sequence has a limit in the set that is used to solve equations and to expand vectors in a basis.
definition [D] (Hilbert Space) A complete inner product space, representing a linear vector space (finite or infinite-dimensional) where every Cauchy sequence of vectors converges to a limit within the space. It serves as the mathematical foundation for the states of quantum systems. A space is a Hilbert space if:
- It possesses a well-defined inner product that induces a norm and a metric.
- It is complete, meaning it has no “holes,” similar to the set of real numbers.
where
- \(H\) is the Hilbert space.
- \(\langle \alpha | \beta \rangle\) is the inner product associating a pair of vectors with a complex scalar.
- \(L^2(a, b)\) is the standard example of an infinite-dimensional Hilbert space consisting of square-integrable functions.
Note:
- \(H\) is also written \(\mathcal{H}\).
140.1 Examples
example 1 [d] (Sequence Space \(\ell^{2}\) — Kreyszig, Shifrin and Adams) As a set,
- \(\ell^{2} = \left\{ x = (\xi_{j})_{j=1}^{\infty} \;\middle|\; \displaystyle\sum_{j=1}^{\infty} |\xi_{j}|^{2} < \infty \right\}\) .
With the inner product
- \((x, y) = \displaystyle\sum_{j=1}^{\infty} \xi_{j}\, \overline{\eta_{j}}\)
and induced norm
- \(\lVert x \rVert = (x, x)^{\dfrac{1}{2}} = \left( \displaystyle\sum_{j=1}^{\infty} |\xi_{j}|^{2} \right)^{\dfrac{1}{2}}\) ,
the space \(\ell^{2}\) is a Hilbert space.
where
- \(\ell^{2}\) is the Hilbert sequence space.
- \(x = (\xi_{j})\) and \(y = (\eta_{j})\) are sequences in \(\ell^{2}\).
- \(\overline{\eta_{j}}\) is the complex conjugate of \(\eta_{j}\).
Note:
- completeness of \(\ell^{2}\) is proved in Kreyszig §1.5-4.
- Shifrin and Adams write \(\ell^{2} = \{ x \in \mathbb{R}^{\omega} : \sum_{k} x_{k}^{2} \text{ exists} \}\) in the real case.
example 2 [d] (Weighted \(L^{2}\) Space — Hassani) As a set, for a strictly positive weight \(w\) on \([a,b]\),
- \(L^{2}_{w}(a,b) = \left\{ f : [a,b] \rightarrow \mathbb{C} \;\middle|\; \displaystyle\int_{a}^{b} |f(x)|^{2}\, w(x)\, dx < \infty \right\}\) .
This is a Hilbert space under the inner product \(\langle g | f \rangle = \int_{a}^{b} g^{*}(x)\, f(x)\, w(x)\, dx\).
where
- \(L^{2}_{w}(a,b)\) is the weighted square-integrable function space on \([a,b]\).
- \(w\) is a strictly positive weight function.
- \(f, g\) are complex-valued functions on \([a,b]\).
- \(g^{*}\) is the complex conjugate of \(g\).
Note:
- when \(w \equiv 1\), this reduces to the usual \(L^{2}(a,b)\).
140.2 Elementary Example
140.2.1 Simple
A Hilbert space is a complete inner product space. The plane \(\mathbb{R}^{2}\) with the dot product is Hilbert.
\[ H = \mathbb{R}^{2} \]
\[ \langle (x_{1},x_{2}), (y_{1},y_{2}) \rangle = x_{1} y_{1} + x_{2} y_{2} \]
\[ \lVert v \rVert = \sqrt{\langle v,v \rangle} \]
where
- \(H\) is the Hilbert space.
- \(\langle \cdot,\cdot \rangle\) is the inner product.
- \(\lVert \cdot \rVert\) is the induced norm.
140.2.2 General
The space \(\ell^{2}\) of square-summable sequences is an infinite-dimensional Hilbert space.
\[ \ell^{2} = \left\{ x = (\xi_{j}) : \sum_{j=1}^{\infty} |\xi_{j}|^{2} < \infty \right\} \]
\[ \langle x,y \rangle = \sum_{j=1}^{\infty} \xi_{j} \overline{\eta_{j}} \]
where
- \(x = (\xi_{j})\) and \(y = (\eta_{j})\) are sequences in \(\ell^{2}\).
- completeness means every Cauchy sequence in \(\ell^{2}\) converges in \(\ell^{2}\).
140.3 References
- Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — \(\ell^{2}\) as sequences with \(\sum|\xi_{j}|^{2}<\infty\); completeness §1.5-4.
- Shifrin, T., & Adams, M. Linear Algebra: A Geometric Approach. W. H. Freeman, 2010. — \(\ell^{2}\) in set-builder notation.
- Hassani, S. Mathematical Physics, 2nd ed. Springer. — \(L^{2}_{w}(a,b)\) as set of square-integrable weighted functions.