Section 4.2 \(L^2[a,b]\)
We saw in the previous section that
\(C[0,1]\) is an inner product space that is not complete. This is unfortunate, as the continuous functions are the foundation of approximation and analysis. The inner product
(2.1.1) is a natural analogue of the inner product in
\(\C^n\text{,}\) considering the extension of sums to integral. It seems a shame to forgo working with the inner product and the space of continuous functions. But if we want to use them, we’ll need to come up with a Hilbert space that contains
\(C[0,1]\text{,}\) which is necessarily going to involve discontinuous functions (see the example at the beginning of the previous section, or the exerise in the problem section below).
The move from \(\C^n\) to \(\ell^2\) provides an analogy - moving to infinite dimensions requires that we use square-summable sequences to get a well-defined inner product. The same turns out to be true here - sums become integrals, so we will work with the square-integrable functions, which we denote \(L^2[0,1]\text{.}\) That is, \(L^2\) consists of those functions \(f\) for which the integral of \(\abs{f}^2\) is finite. The sharp-eyed reader may notice that we have not written this in mathematics as \(\int_0^1 \abs{f}^2 \, \dd t\) and instead used the weasel-word “integrable”.
At issue is how we define the integral. The Riemann integral is the familiar \(\int f \, \dd x\) that we learn in elementary calculus. Many more functions \(f\) on \([0,1]\) will satisfy
\begin{equation*}
\int_0^1 \abs{f}^2 \dd t \lt \infty,
\end{equation*}
than
\(C[0,1]\text{,}\) and yet even this space is incomplete. (Consider the sequence of Riemann integrable functions
\(f_n\) on
\([0,1]\) attained recursively by introducing an additional discontinuity at the
\(n\)th rational number in
\([0,1]\) under some enumeration of
\(\mathbb{Q}\text{.}\) The limit function is the Dirichlet function
\(s(t)\) defined below.) We need a better integral. The
Lesbesgue integral is significantly more challenging to develop than the Riemann integral, and we will not attempt that in this course. It is enough for us to accept that a more powerful notion of integration exists so that the space of square-integrable functions is complete with respect to
(2.1.1).
Example 4.2.1. \(L^2(a,b)\).
Let \(-\infty\leq a \lt b \leq \infty\text{.}\) \(L^2(a,b)\) is the space of Lesbesgue measureable functions \(f:(a,b)\to \C\) which are square-integrable - that is,
\begin{equation*}
\int_a^b \abs{f}^2 \, \dd t \infty,
\end{equation*}
with pointwise operations and inner product
(2.1.1).
The condition that
\(f\) be Lesbesgue measurable is not terribly restrictive. One can study and use the theory of
\(L^2\) functions in a wide variety of physical and mathematical applications without ever needing to grapple with the oddities of non-measureable functions, which are quite difficult to construct. To get an idea of how bad a function can be and still be Lesbesgue integrable, consider the
Dirichlet function
\begin{equation*}
s(t) = \left\{
\begin{array}{ll}
1 \amp \text{ if } t \in [0,1]-\mathbb{Q} \\
0 \amp \text{ if } t \in [0,1]\cap \mathbb{Q}
\end{array}
\right.
\end{equation*}
which integrates to 1 in the Lesbesgue integral, while the Riemann integral runs screaming in horror.
There is one issue that needs to be addressed when defining membership in a space with an integral condition - when should we consider two functions to be equal? If we take the function \(f(t) = t\) and move one point, say \(g(t) = t\) when \(t \neq \frac{1}{2}\) and \(g(t) = 0\) when \(t = \frac{1}{2}\text{,}\) the functions behave identically under the integral and thus in the inner product and the metric in \(L^2\text{.}\) More precisely, if two functions differ only at a finite set of points in \((a,b)\text{,}\) then
\begin{equation*}
\norm{f - g}^2 = \int_a^b \abs{f - g}^2 \, \dd t = 0
\end{equation*}
and so by
Definition 3.1.3 (1),
\(f\) and
\(g\) must be equal in
\(L^2\text{.}\)
So how much can two functions differ and still have the same integral? We define a subset \(N\) of \(\R\) to be a null set if for any \(\eps > 0\) there exists a sequence of intervals \((a_n, b_n)\) with \(E \subseteq \cup_n (a_n, b_n)\) and \(\sum_n (b_n - a_n) \lt \eps\) - that is, \(E\) is a null set if it is contained in the union of a set of intervals of arbitrarily small total length. Two functions are said to be equal almost everywhere if they differ only on a null set. In the Lesbesue integral, if \(f\) and \(g\) are equal almost everywhere, then the integral of \(\abs{f - g}^2\) is zero, and so \(f\) and \(g\) must be regarded as equal in \(L^2\text{.}\) This indicates that the elements of \(L^2\) are properly considered as equivalence classes of almost everywhere equal functions. In practice, we still refer to the elements as functions and treat equality as almost everywhere equality. This viewpoint does, however, mean that functions need only be defined almost everywhere to belong to \(L^2\text{.}\) For example, \(f(t) = \frac{1}{\sqrt[4]{\abs{t}}}\) is in \(L^2[-1,1]\) despite the asymptote at \(t = 0\text{.}\)
One of the more useful properties of \(L^2[a,b]\) is that the continuous functions \(C[a,b]\) are a dense subspace (that is, every \(L^2\) function can be approximated by a sequence of continuous functions). A typical approach in analysis is to show that some property hold on the continuous functions and then extend the result to \(L^2\) by approximation.
Theorem 4.2.2.
\(L^2[a,b]\) is a Hilbert space.
Theorem 4.2.3.
\(C[a,b]\) is a dense subspace of \(L^2[a,b]\text{.}\)
The theorem above is a consequence of measure theory, which requires the development of the Lebesgue integral to justify. We must accept the result, for the moment, on faith. The interested reader should look at
Lusin’s Theorem and the
Tietze Extension Theorem for a taste of a standard method of proof in the general case.
