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Advanced Calculus, part I Analysis and Topology in Normed Spaces

Andrew Cooper

Section 3.2 distances

A fundamental notion in our interpretation of limits is distance. But since we have size and subtraction, we get distance automatically.
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Subsection 3.2.1 the induced metric

Definition 3.2.1.

The metric induced by a norm \(\lVert\cdot\rVert\) on a vector space \(V\) is the function \(d:V\times V\to [0,\infty)\) given by
\begin{equation*} d(x,y)=\lVert x-y\rVert\ \ . \end{equation*}
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Example 3.2.2.

Each \(p\)-norm on \(\mathbb{R}^m\) induces a \(p\)-metric on \(\mathbb{R}^m\text{:}\)
\begin{equation*} d_p(x,y)=\lVert x-y\rVert_p \end{equation*}
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Checkpoint 3.2.3.

Compute the distances among the following points in \(\mathbb{R}^2\text{,}\) using the indicated \(p\)-metric:
  1. \(\displaystyle d_1\left((1,1),(2,3)\right)\)
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  2. \(\displaystyle d_2\left((1,1),(2,3)\right)\)
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  3. \(\displaystyle d_\infty\left((1,1),(2,3)\right)\)
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Remark 3.2.4.

The \(1\)-metric \(d_1\) on \(\mathbb{R}^2\) is sometimes called the taxicab metric.
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Proof.

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Subsection 3.2.2 metric spaces

Most of what we’re going to talk about in terms of topology will only require the metric itself, and it won’t be necessary to fall back on computing norms.
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Definition 3.2.6.

A metric space \((S,d)\) is a set \(S\text{,}\) together with a function \(d:S\times S\to [0,\infty)\) so that:
S
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for any \(x,y\in S\text{,}\) \(d(x,y)=d(y,x)\)
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\(d(x,y)\geq 0\text{;}\) we have equality if and only if \(x=y\)
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for any \(x,y,z\in S\text{,}\) \(d(x,z)\leq d(x,y)+d(y,z)\)
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Example 3.2.9.

Take \(V=\mathbb{R}^3\text{,}\) \(\lVert\cdot\rVert_2\text{,}\) and \(S=\left\{(x,y,z)\in\mathbb{R}^3\middle\vert x^2+y^2+z^2=1\right\}\text{.}\)
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Checkpoint 3.2.10.

Which definitions and theorems from Chapter 2 make sense in a metric space? For any that do, formulate them.
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Subsection 3.2.3 from intervals to balls

It’s going to turn out that the shapes of sets are important to understanding functions. (The word topology means, roughly, the mathematical study of shapes -- which is why it appears as the title of this chapter.)
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There aren’t all that many interestingly-shaped sets in \(\mathbb{R}\text{.}\) The most basic kind of subsets of \(\mathbb{R}\) are the intervals, which come in two flavors: open and closed.
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Observe that each interval has a midpoint and a length; although we normally describe an interval like
\begin{equation*} (3,16) \end{equation*}
as "the open interval from 3 to 17", we could as well specify it as "the open interval whose midpoint is 10 and whose length is 14", or perhaps as "the open interval centered at 10 with radius 7".
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In a normed space (or a metric space), we don’t have any notion of order, so intervals don’t make sense. But we can certainly talk about distances and centers. So we’re going to generalize the idea of an interval like this:
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Definition 3.2.11.

The open ball with center \(x\) and radius \(r\) is the set
\begin{equation*} B_r(x)=\left\{y\ \middle\vert\ d(x,y)\lt r\right\} \end{equation*}
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Definition 3.2.12.

The closed ball with center \(x\) and radius \(r\) is the set
\begin{equation*} D_r(x)=\left\{y\ \middle\vert\ d(x,y)\leq r\right\} \end{equation*}
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Remark 3.2.13.

Observe that because the definitions of open and closed balls are framed solely in terms of the metric function, they make sense in any metric space.
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Checkpoint 3.2.14.

Convince yourself that if we think of \(\mathbb{R}\) as a normed space with \(\lvert\cdot\rvert\text{,}\) then the open balls in \(\mathbb{R}\) are precisely the open intervals, and the closed balls are precisely the closed intervals.
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Checkpoint 3.2.15.

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