Modes of convergence

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Section 5.3 Modes of convergence

A point that we need to pay attention to in more abstract spaces than \(\R^n\) is precisely what we mean by convegence. In the previous section, we discussed convergence in \(L^2(a,b)\) - i.e. given a sequence \(f_n\) in \(L^2(0,1)\text{,}\) what it means for

\begin{equation*} \sum_{n=1}^\infty f_n = f. \end{equation*}

Because this mode of \(L^2\)-convergence is in norm and incorporates information from the entire domain, we cannot conclude that for a given \(t\text{,}\) the series

\begin{equation*} \sum_{n=1}^\infty f_n(t) \to f(t). \end{equation*}

Series that satisfy the above notion are said to converge pointwise, and these two ideas are not equivalent. In fact, a great deal of the nuance of graduate real analysis is in exploring the differences between these and other modes of convergence, usually in the guise of measure theory (here, an extremely well presented introduction can be found in S. Axler’ Measure, Integration, and Real Analysis or G. Folland’s Real Analysis: Modern Techniques and Their Applications).

Because series convergence is really just convergence of the sequence of partial sums in the senses that we are discussing, we’ll look at the difference in convergence modes for several sequences. First, consider the sequence of functions \(g_n \in L^2[0,1)\) given by

\begin{equation*} g_n(t) = \sqrt{n}t^n. \end{equation*}

Checkpoint 5.3.1.

Show that for all \(t \in [0,1)\text{,}\) \(g_n(t) \to 0\text{.}\)

Show that the sequence \(\norm{g_n - 0}^2_{L^2}\) does not converge to \(0\text{.}\)

Conclude that pointwise convergence does not imply \(L^2\) convergence.

Now consider the sequence of functions given by

\begin{equation*} h_n(t) = \left\{ \begin{array}{ll} \sqrt[3]{n} \amp\mbox{ for } t \in [0,\frac{1}{n}] \\ 0 \amp\mbox{ otherwise}\end{array}\right. . \end{equation*}

Checkpoint 5.3.2.

Sketch a graph of \(h_n\text{.}\)

Show that the sequence \(h_n(0)\) does not converge.

Show that \(\norm{h_n - 0}^2 \to 0\text{,}\) and conclude that \(h_n \to 0\) in \(L^2.\)

Conclude that \(L^2\) convergence does not imply pointwise convergence.

The upshot of this discussion is: be cautious. The same problems beset the great mathematicians of the 19th century, so let us strive to learn from their struggles.