So I was flipping through Lang's(adsbygoogle = window.adsbygoogle || []).push({}); Undergraduate Analysisand noticed the absence of the important concept of metric spaces. I checked the index and was referred to problem 2, Chapter 6 Section 2. There he defines a metric space, what a bounded metric is, and gives a few straightforward problems about specific bounded metrics. This is not very interesting, but the next problem is.

In fact, I'll just copy the problem here for your convenience. (FYI, I have 'solved' the problem so I'm not asking for help.)

Problem 3:

Let S be a metric space. For each x in S, define a function [itex] f_x : S \rightarrow \mathbb{R} [/itex] by the formula [itex] f_x(y) = d(x,y) [/itex].

(a) Given two points x, a in S, show that [itex] f_x - f_a [/itex] is a bounded function on S.

(b) Show that [itex] d(x,y) = \left|\left|f_x - f_y \right|\right| [/itex]

(c) Fix an element a of S. Let [itex] g_x = f_x - f_a [/itex]. Show that the bounded map [itex] x \mapsto g_x [/itex] is a distance preserving embedding (i.e., injection) of S into the normed vector space of bounded functions on S with the sup norm. [If the metric on S was orginally bounded, you can use f_x instead of g_x.] This exercise shows that the generality of metric spaces is illusory. In applications, metric spaces arise naturally as subsets of normed vector spaces.

So there it is. I don't think metric spaces are ever mentioned again in the book, although admittedly I have not carefully read the entire book looking for their mention. Lang develops the rest of the standard topics in undergraduate analysis in the context of normed linear spaces (and of course, after the notion of completeness has been expounded, most of the analysis takes place in Banach spaces, although he does not use that word, for whatever reason.)

Now like most of you, I first learned about continuity, completeness, compactness, etc., in the context in metric spaces. Then, upon reaching normed linear spaces, which are obviously also metric spaces, we could just 'transfer' over all the results from metric space theory. After reaching normed linear spaces, metric spaces are really never mentioned again, except maybe to quote a result from their theory.

So why the focus in standard undergraduate analysis textbooks on metric spaces when (a) as Lang, shows, their generality is illusory and (b) we are really after normed linear space theory?

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# The illusory generality of metric spaces

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