Chapter 10

If \(A\) is a dense set in a topological space \(X\) and \(U \subseteq X\) is open, show that \(U \subseteq \overline{A \cap U}\).

Show that a map \(f \cdot X \rightarrow Y\) between two topological spaces \(X\) and \(Y\) is contintous if and only if \(f(\bar{U}) \subseteq \overline{f(U)}\) for all sets \(U \subseteq X\) Show that \(f\) is a lwomeomorplism only if \(f(\bar{U})=\overline{f(U)}\) for all sets \(U \subseteq X\)

If \(W, X\) and \(Y\) are topologieal spaces and the functions \(f: W \rightarrow X, g \quad X \rightarrow Y\) are toth contunuous, show that the function \(h=g \circ f, W \rightarrow Y\) is contimous.

A topological space \(X\) is called normal if for evcry pair of disjo?nt closcd suhsets \(A\) and \(B\) there exist disjoint open sets \(U\) and \(V\) sich that \(A \subset U\) and \(B \subset V\) Show that every metric space is nomal

If \(f: X \rightarrow Y\) is a continuous map betwecn lopological spaces, we define its graph to be the set \(G=\\{(x, f(x)) \mid x \in, X] \subseteq X \times Y\). Show that if \(G\) is given the relative topology induced by the topological product \(X \times Y\) then it is homeonorphic to the lopological space \(X\)

Let \(X\) and \(Y\) be topological spaccs and \(f . X \times Y \rightarrow X\) a continuous map. For each fixed \(a \in X\) show that the map \(f_{n}: Y \rightarrow X\) defined by \(f_{\theta}(v)=f(a, v)\) is contunous.

Show that ff \(f . X \rightarrow Y\) and \(g \cdot X \rightarrow Y\) are continuous maps from a topological space \(X\) into a Haucdorff space \(Y\) then the set of pomts \(A\) on whel these maps agrec, \(A=\\{x \in\) \(X \mid f(\mathrm{r})=g(x)\\}\), s closed. If \(A\) is a dense subset of \(X\) show that \(f=g\)

If \(G_{0}\) is the component of the identity of a locally connected topological group \(G\), the factor group \(G / G_{0}\) is called the group of components of \(G .\) Show that the group of components is a discrete topological group with respect to the topology induced by the natural projection map \(\pi: g \mapsto g G_{0}\)

Show that a linear map \(T: V \rightarrow W\) between topological vector spaces is continuous everywhere on \(V\) if and only if it is continuous at the origin \(0 \in V\).

Show that the following are all norms in the vector space \(\mathbb{R}^{2}\) : $$ \begin{aligned} &\|\mathbf{u}\|_{1}=\sqrt{\left(u_{1}\right)^{2}+\left(u_{2}\right)^{2}} \\ &\|\mathbf{u}\|_{2}=\max \left[\left|u_{1}\right|,\left|u_{2}\right|\right\\} \\\ &\|\mathbf{u}\|_{3}=\left|u_{1}\right|+\left|u_{2}\right| \end{aligned} $$ What are the shapes of the open balls \(B_{a}(\mathrm{u})\) ? Show that the topologes generated by these norms are the same.

Show that if \(x_{n} \rightarrow x\) in a normed vector space then $$ \frac{x_{1}+x_{2}+\cdots+x_{n}}{n} \rightarrow x $$

Show that if \(x_{n}\) is a sequence in anormed vector space \(V\) such that every subsequence has a subsequence comergent to \(x\), then \(x_{n} \rightarrow x\).

Let \(V\) be a Banach space and \(W\) te a vector subspace of \(V .\) Define its closture \(\bar{W}\) to be the union of \(W\) and all hmuts of Cauchy sequences of elements of \(W\). Show that \(\bar{W}\) is a closed vector subspace of \(V\) in the sense that the limit points of all Cauchy scquences in \(\bar{W}\) lie in \(\bar{W}\) (note that the Cauchy sequences may include the newly added limit points of \(W\) ).

Show that cvery space \(F(S)\) is complete with respect to the supremum norm of Example 10.26. Hence show that the vector space \(\ell_{\infty}\) of bounded infinite complex sequences is a Banach space with respect to the norm \(\|\mathrm{x}\|=\sup \left(x_{t}\right)\).

Show that the set \(V^{\prime}\) consisting of bounded linear functionals on a Banach space \(V\) is a normed vector space with respect to the norm $$ \|\varphi\|=\sup [M|| \varphi(x) \mid \leq M\|x\| \text { for all } x \in V \mid $$ Show that this norm is complete on \(V^{\prime}\).

We say two norms \(\|u\|_{1}\) and \(\|u\|_{2}\) on a vector space \(V\) are equivalent if there exist constants \(A\) and \(B\) such that $$ \|u\|_{1} \leq A\|u\|_{2} \quad \text { and }\|u\|_{2} \leq B\|u\|_{1} $$ for all \(u \in V\). If two norms are equivalent then show the following: (a) If \(u_{n} \rightarrow u\) with respect to one norm then this is also true for the other norm. (b) Every linear functional that is continuous with respect to one norm is continuous with respect to the other norm. (c) Let \(V=C[0,1]\) be the vector space of continuous complex functions on the interval \([0,1]\). By considering the sequence of functions $$ f_{n}(x)=\frac{n}{\sqrt{\pi}} \mathrm{e}^{-m^{2} x^{2}} $$ show that the norms $$ \|f\|_{1}=\sqrt{\int_{0}^{1}|f|^{2} \mathrm{~d} x} \text { and }\|f\|_{2}=\max \\{f(x)|| 0 \leq x \leq 1\\} $$ are not equivalent. (d) Show that the linear functional defined by \(F(f)=f(1)\) is contunuous wath respect to \(\|\cdot\|_{2}\) but not with respect to \(\|\cdot\|_{1}\).