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.