\begin{equation*}
\begin{array}{l|l|l}
\text{Space} \amp \text{Elements} \amp \text{Inner product} \\ \hline
\C^n \amp x = (x_1, \ldots, x_n), \, x_i \in \C \amp \ip{x}{y} = \sum x_i \cc{y}_i\\
\C^{m\times n} \amp x = (x_{ij})_{i = 1, j=1}^{m, n}, \, x_{ij} \in \C \amp \ip{A}{B} = \trace(B\ad A)\\
\ell^2 \amp x = (x_i)_{i=1}^\infty, \, x_i \in \C, \, \sum \abs{x_i}^2 \lt \infty \amp \ip{x}{y} = \sum_1^\infty x_i \cc{y_i} \\
C[a, b] \amp f:[a,b] \to \C \text{ continuous} \amp \ip{f}{g} = \int_a^b f(t) \cc{g(t)} \, \dd t \\
RL^2 \amp f \text{ rational, analytic on } \T \amp \ip{f}{g} = \frac{1}{2 \pi i} \int_\T f(z) \cc{g(z)}\, \frac{\dd z}{z} \\
RH^2 \amp f \text{ rational, analytic on } \cl \D \amp \ip{f}{g} = \frac{1}{2 \pi i} \int_\T f(z) \cc{g(z)}\, \frac{\dd z}{z} \\
TP \amp f(x) = \sum_{-N}^N c_n e^{inx} \amp \ip{f}{g} = \frac{1}{2\pi} \int_{-T}^T f(x)\cc{g(x)}\, \dd x
\end{array}
\end{equation*}
Section 2.3 Examples of inner product spaces
So far, our examples of inner product spaces have been:
- \(\C^n\text{:}\) the space of \(n\) dimensional complex vectors equipped with the inner product\begin{equation*} \ip{x}{y} = \sum_{i=1}^n \cc{y}_i x_i\text{;} \end{equation*}
- \(\ell^2\text{:}\) the space of square summable infinite sequences equipped with the inner product\begin{equation*} \ip{x}{y} = \sum_{i=1}^\infty \cc{y_i} x_i; \end{equation*}
- \(\C^{m\times n}\text{:}\) the space of \(m \times n\) complex matrices equipped with the inner product\begin{equation*} \ip{A}{B} = \trace (B\ad A); \end{equation*}
- \(C[0,1]\text{:}\) the space of complex-valued continuous functions on the interval \([0,1]\) equipped with the inner product\begin{equation*} \ip{f}{g} = \int_0^1 f(t) \cc{g(t)} \, \dd t; \end{equation*}
To develop intuition and to show how inner product gemetry interacts with other mathematical structures, particularly those of analysis, we need more examples. The first three spaces in some sense very much resemble Euclidean space. They are easy to work with but don’t reveal the more nuanced structure of Hilbert spaces. On the other hand, general continuous functions are about as hard to grasp a collection of objects as exists in mathematics (“I turn away with fear and horrow from the lamentable plague of continuous functions which do not have derivatives...” Hermite to Stieltjes, 1893 - unfortunately, most continuous functions are pathological in this sense).
1
nautil.us/issue/11/light/maths-beautiful-monsters2
homepages.math.uic.edu/~marker/math414/fs.pdfExamples of inner product spaces with non-trivial geometry include spaces of rational functions (that is, quotients of complex polynomials), which provide interesting examples and turn out to be of importance in applications. Rational functions also provide a foundation for the study of functions evaluated on matrix inputs, which will be considered much later in this text.
It is most natural to consider the following examples in the context of complex-valued functions. A function \(f:D \to \C\) is analytic if \(f\) has a power series that converges in a neighborhood of every \(x_0 \in D\text{.}\) Denote by \(RL^2\) the space of rational functions that are analytic on the unit circle \(\T = \partial \D = \{z\in \C: \abs{z} = 1\},\) equipped with pointwise addition and scalar multiplication. \(RL^2\) is an inner product space with the inner product
3
en.wikipedia.org/wiki/Analytic_function
\begin{equation}
\ip{f}{g} = \frac{1}{2\pi i} \int_\T f(z) \cc{g(z)} \frac{\dd z}{z},\tag{2.3.1}
\end{equation}
with the integral taken as \(z\) travels around \(\T\) counterclockwise. \(RL^2\) is those rational functions with no pole on the unit circle \(\T\text{.}\)
4
en.wikipedia.org/wiki/Zeros_and_polesWe can restrict \(RL^2\) to the subspace of functions that are analytic on the closed unit disk \(\cl{\D}\text{,}\) where \(\D = \{z: \abs{z} \lt 1\}\text{.}\) We will denote this space of functions \(RH^2\text{.}\) It is an inner product space with the same inner product as \(RL^2\text{.}\) These spaces are subspaces of the larger spaces of functions \(L^2\) and \(H^2\text{,}\) which will be discussed detail as we proceed.
Another family of important function spaces are called Sobolev spaces. The simplest example is the space \(W[a,b]\) of continuously differentiable functions with values in \(\C\) with the inner product
\begin{equation}
\ip{f}{g} = \int_a^b f(t) \cc{g(t)} + f'(t) \cc{g'(t)} \, \dd t.\tag{2.3.2}
\end{equation}
Including the derivative in the inner product gives control over the behavior of both the function and the derivative. \(W\) and the broader family of Sobolev spaces are important in the study of differential equations.
Of central importance in engineering and the sciences is the space of trigonometric polynomials, which are functions of the form
\begin{equation*}
T(x) = a_0 + \sum_{n=1}^N a_n \cos(nx) + i\sum_{n=1}^N b_n \sin nx,
\end{equation*}
where \(a_n, b_n \in \C\) and \(1 \leq n \leq N\text{.}\) Using Euler’s identity, it is computationally efficient to use the equivalent characterization
\begin{equation*}
f(x) = \sum_{n=-N}^N c_n e^{inx}.
\end{equation*}
The space \(TP\) of trigonometric polynomials is a complex vector space with pointwise addition and scalar multiplication. That the relation
\begin{equation}
\ip{f}{g} = \frac{1}{2\pi} \int_{-T}^T f(x) \cc{g(x)} \, \dd x\tag{2.3.3}
\end{equation}
is an an inner product on \(TP\) is left as an exercise.
