So, we know that \(PC(a,b)\) is incomplete. This means that it is “missing” the limits of some sequences of functions.
We search for a bigger space of functions containing \(PC(a,b)\).
This leads us to:
Definition
We write \(L^2(a,b)\) for the space of all integrable functions \(f:[a,b] \to \mathbb{C}\) for which
\[ \int_a^b |f(x)|^2 \, dx < \infty. \tag{1} \]
Notice that there are no continuity or smoothness requirements for \(f\) to be in \(L^2(a,b)\). The only requirements are:
That (1) holds.
That \(f\) is measurable. (This is a technical requirement that is covered in more advanced classes.)
Related to the “measurability” requirement is that the integral in (1) is technically a Lebesgue integral, which is an extension of the familiar Riemann integral. Again, this is covered in more advanced classes.
Theorem
The space \(L^2(a,b)\) contains \(PC(a,b)\).
The inner product and norm on \(PC(a,b)\) extend to an inner product and norm on \(L^2(a,b)\).
The space \(L^2(a,b)\) is complete.
Theorem 3.1
Let \(\{\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_k\}\) be an orthonormal set of vectors in \(\mathbb{C}^k\). Then for any \(\mathbf{a} \in \mathbb{C}^k\), we have
\[ \mathbf{a} = \langle \mathbf{a}, \mathbf{u}_1 \rangle \mathbf{u}_1 + \langle \mathbf{a}, \mathbf{u}_2 \rangle \mathbf{u}_2 + \cdots + \langle \mathbf{a}, \mathbf{u}_k \rangle \mathbf{u}_k, \]
and \[ \| \mathbf{a} \|^2 = |\langle \mathbf{a}, \mathbf{u}_1 \rangle|^2 + |\langle \mathbf{a}, \mathbf{u}_2 \rangle|^2 + \cdots + |\langle \mathbf{a}, \mathbf{u}_k \rangle|^2. \]
We asked: Is there an analog of this theorem for spaces of functions? We will see that the answer is yes for \(L^2(a,b)\).
This will involve an infinite orthonormal set \(\{\phi_n\}_{n=1}^\infty\) of functions, and the expansion will look like \[ f = \sum_{n=1}^\infty \langle f, \phi_n \rangle \phi_n. \]
We have to do two things: Establish that the series on the right converges (in what mode?) and that the series converges to \(f\).
Bessel’s Inequality
If \(\{\phi_n\}_{n=1}^\infty\) is an orthonormal set in \(L^2(a,b)\) and \(f\in L^2(a,b)\), then
\[ \sum_{n=1}^\infty |\langle f,\phi_n \rangle|^2 \leq \|f\|^2. \]
Lemma 3.2
If \(f\) is in \(L^2(a,b)\) and \(\{\phi_n\}_{n=1}^\infty\) is any orthonormal set in \(L^2(a,b)\), then the series \(\sum_{n=1}^\infty \langle f,\phi_n \rangle \phi_n\) converges in norm and
\[ \left\| \sum_{n=1}^\infty \langle f, \phi_n \rangle \phi_n \right\| \leq \|f\|. \]
Theorem 3.4
Let \(\{\phi_n\}_{n=1}^\infty\) be an orthonormal set in \(L^2(a,b)\). The following statements are equivalent:
If \(\langle f,\phi_n\rangle =0\) for all \(n\), then \(f=0\).
For every \(f\) in \(L^2(a,b)\), we have \(f = \sum_{n=1}^\infty \langle f, \phi_n \rangle \phi_n\) in norm.
For every \(f\) in \(L^2(a,b)\), we have \[ \|f\|^2 = \sum_{n=1}^\infty |\langle f, \phi_n \rangle|^2. \]
Definition
If an orthonormal set \(\{\phi_n\}_{n=1}^\infty\) in \(L^2(a,b)\) has the (equivalent) properties listed in Theorem 3.4, then it is called a complete orthonormal set or an orthonormal basis.
More generally, suppose that \(\{\psi_n\}_{n=1}^\infty\) is an orthogonal set in \(L^2(a,b)\). We shall say that this set is a complete orthogonal set or an orthogonal basis if the set \(\{\phi_n\}_{n=1}^\infty\) with \(\phi_n = \psi_n / \|\psi_n\|\) is an orthonormal basis.
Theorem 3.5
The set \(\{e^{inx}\}_{n=-\infty}^\infty\) is an orthogonal basis of \(L^2(-\pi,\pi)\).
Given \(f\in L^2(a,b)\), explain why the Fourier series of \(f\) converges to \(f\) in norm.
Note
Let \(f:[-\pi,\pi] \to \mathbb{R}\) be an integrable function, and consider its Fourier series.
If \(f\) is continuous and piecewise smooth, then the Fourier series converges to \(f\) uniformly, pointwise, and in norm.
If \(f\) is piecewise smooth, then the Fourier series converges to \(f\) pointwise and in norm.
If \(f\) is in \(L^2(-\pi,\pi)\), then the Fourier series converges to \(f\) in norm.