Problem 12
Prove that if two functions \(f, g\) with values in a normed vèctor space are uniformly continuous on a set \(B,\) so also are \(f \pm g\) and \(a f\) for a fixed scalar \(a\). For real functions, prove this also for \(f \vee g\) and \(f \wedge g\) defined by $$(f \vee g)(x)=\max (f(x), g(x))$$ and $$ (f \wedge g)(x)=\min (f(x), g(x)) $$
Problem 13
Prove that if \(\left\\{f_{n}\right\\}\) tends to \(f\) (pointwise or uniformly), so does each subsequence \(\left\\{f_{n_{k}}\right\\}\).
Problem 13
A set is said to be totally disconnected iff its only connected subsets are one-point sets and \(\emptyset\). Show that \(R\) (the rationals) has this property in \(E^{1}\).
Problem 14
Prove that every compact set is complete. Disprove the converse by examples.
Problem 14
In the following cases, show that \(f\) is uniformly continuous on \(B \subseteq
E^{1}\), but only continuous (in the ordinary sense) on \(D,\) as indicated, with
\(0
Problem 15
Prove that if \(f\) is uniformly continuous on \(B,\) it is so on each subset \(A \subseteq B\).
Problem 15
Prove that \(\left\\{\left(x_{m}, y_{m}\right)\right\\}\) is a Cauchy sequence in \(X \times Y\) iff \(\left\\{x_{m}\right\\}\) and \(\left\\{y_{m}\right\\}\) are Cauchy. Deduce that \(X \times Y\) is complete iff \(X\) and \(Y\) are.
Problem 17
Prove that the convergence or divergence (pointwise or uniformly) of a sequence \(\left\\{f_{m}\right\\}\), or a series \(\sum f_{m}\), of functions is not affected by deleting or adding a finite number of terms. Prove also that \(\lim _{m \rightarrow \infty} f_{m}\) (if any) remains the same, but \(\sum_{m=1}^{\infty} f_{m}\) is altered by the difference between the added and deleted terms.
Problem 18
Let \(\bar{p}_{0}, \bar{p}_{1}, \ldots, \bar{p}_{m}\) be fixed points in \(E^{n}\left({ }^{*}\right.\) or in another normed space \()\). Let $$f(t)=\bar{p}_{k}+(t-k)\left(\bar{p}_{k+1}-\bar{p}_{k}\right)$$ whenever \(k \leq t \leq k+1, t \in E^{1}, k=0,1, \ldots, m-1\) Show that this defines a uniformly continuous mapping \(f\) of the interval \([0, m] \subseteq E^{1}\) onto the "polygon" $$\bigcup_{k=0}^{m-1} L\left[p_{k}, p_{k+1}\right]$$ In what case is \(f\) one to one? Is \(f^{-1}\) uniformly continuous on each \(L\left[p_{k}, p_{k+1}\right] ?\) On the entire polygon?
Problem 19
Prove the sequential criterion for uniform continuity: A function \(f: A \rightarrow T\) is uniformly continuous on a set \(B \subseteq A\) iff for any two (not necessarily convergent) sequences \(\left\\{x_{m}\right\\}\) and \(\left\\{y_{m}\right\\}\) in \(B,\) with \(\rho\left(x_{m}, y_{m}\right) \rightarrow 0,\) we have \(\rho^{\prime}\left(f\left(x_{m}\right), f\left(y_{m}\right)\right) \rightarrow 0\) (i.e., \(f\) preserves concurrent pairs of sequences; see Problem 4 in Chapter \(3, \S 17 ).\)
