Introductory university-level calculus, linear algebra, abstract algebra, probability, statistics, and stochastic processes.
Vector space of functions
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We study the collection of functions \(\mathcal{R}_{2\pi}^2\) (square-integrable functions with period \(2\pi\)) as a vector space. We define the \(L^2\) norm and \(L^2\) inner product on this vector space so that we can investigate the ``geometric'' structure of the space of functions.
Let us show that \(\mathcal{R}_{2\pi}^{2}\), the set of square-integrable functions with period \(2\pi\), is a vector space over \(\mathbb{C}\). First, we need to define addition and scalar multiplication. Let \(f, g\in \mathcal{R}_{2\pi}^2\). We define \(f + g \in \mathcal{R}_{2\pi}^2\) by
\[(f+g)(x) = f(x) + g(x), ~ x \in \mathbb{R}.\tag{eq:add}\]
Note that the ``\(+\)'' on the left-hand side is defined between the two functions \(f\) and \(g\), whereas the ``\(+\)'' on the right-hand side is the addition between two complex numbers \(f(x)\) and \(g(x)\). Next, we define scalar multiplication. Let \(\alpha\in \mathbb{C}\) and \(f \in \mathcal{R}_{2\pi}^2\). We define \(\alpha f\) by
\[(\alpha f)(x) = \alpha f(x).\tag{eq:scale}\]
Note that the product on the left-hand side is between the scalar \(\alpha\) and the function \(f\), whereas the product on the right-hand side is between the two complex numbers \(\alpha\) and \(f(x)\).
Lemma
\(\mathcal{R}_{2\pi}^2\) is a vector space with vector addition (eq:add) and scalar multiplication (eq:scale).
Proof. We show that \(\mathcal{R}_{2\pi}^2\) satisfies all the axioms of the vector space. In the following, \(f, g, h\in\mathcal{R}_{2\pi}^2\) and \(\alpha, \beta \in \mathbb{C}\).
1. \(\mathcal{R}_{2\pi}^2\) is closed under vector addition.
Suppose \(f, g \in \mathcal{R}_{2\pi}^2\). We show \(f + g\in\mathcal{R}_{2\pi}^2\).
By assumption, \(\int_{\pi}^{\pi}|f(x)|^2\,dx < +\infty\) and \(\int_{-\pi}^{\pi}|g(x)|^2\,dx < +\infty\). Thus,
Thus, \(\mathcal{R}_{2\pi}^{2}\) is a vector space over \(\mathbb{C}\). ■
Definition (\(L^2\)-norm, mean-square norm)
Let \(f\) be a function on \((-\pi, \pi)\) that is square-integrable, i.e.,
\[\int_{-\pi}^{\pi}|f(x)|^2dx < +\infty.\]
We define the mean-square norm or \(L^2\) norm \(\|f\|\) by
\[\|f\| = \sqrt{\int_{-\pi}^{\pi}|f(x)|^2dx}.\]
Remark. The \(L\) in \(L^2\) stands for ``Lebesgue.'' suggesting that we should use the Lebesgue integral rather than the Riemann integral. But the term ``\(L^2\)'' is so widespread that we use it, although we only use the Riemann integral. □
Remark. The \(L^2\) inner product corresponds to the scalar (dot) product in a vector space. □
In general, an inner product is defined as follows.
Definition (Inner product (general))
Let \(V\) be a vector space over the field \(K\). An inner product \((\cdot, \cdot): V\times V \to K\) is a map with the following properties for all vectors \(x,y,z\in V\) and all scalars \(\alpha, \beta \in K\):
(Linearity in the first argument) \[(\alpha x + \beta y, z) = \alpha(x,z) + \beta(y,z)\]
(Positive definiteness) If \(x \neq 0\), \[(x,x) > 0.\]
Example. Consider the vector space \(\mathbb{R}^n\). For \(x = (x_1, x_2, \cdots, x_n), y = (y_1, y_2, \cdots, y_n) \in \mathbb{R}^n\), the scalar product is defined as
\[(x,y) = \sum_{i=1}^{n}x_iy_i.\]
For each \(a \in \mathbb{R}\), its ``conjugate'' is the same \(a\): \(\overline{a} = a\). Thus,
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