Chapter 7

Finish the proof of Proposition 7.2.11. Suppose \((X, d)\) is a metric space and \(Y \subset X .\) Show that with the subspace metric on \(Y\), if a set \(U \subset Y\) is open \((\) in \(Y)\), then there exists an open set \(V \subset X\) such that \(U=V \cap Y\)

Let \((X, d)\) be a complete metric space. Show that \(K \subset X\) is compact if and only if \(K\) is closed and such that for every \(\varepsilon>0\) there exists a finite set of points \(x_{1}, x_{2}, \ldots, x_{n}\) with \(K \subset \bigcup_{j=1}^{n} B\left(x_{j}, \varepsilon\right) .\) Note: Such a set \(K\) is said to be totally bounded, so in a complete metric space a set is compact if and only if it is closed and totally bounded.

Let \(C^{1}([a, b], \mathbb{R})\) be the set of once continuously differentiable functions on \([a, b] .\) Define $$d(f, g):=\|f-g\|_{u}+\left\|f^{\prime}-g^{\prime}\right\|_{u}$$ where \(\|\cdot\|_{u}\) is the uniform norm. Prove that \(d\) is a metric.

Let \((X, d)\) be a metric space. a) For any \(x \in X\) and \(\delta>0,\) show \(\overline{B(x, \delta)} \subset C(x, \delta)\) b) Is it always true that \(\overline{B(x, \delta)}=C(x, \delta) ?\) Prove or find a counterexample.

Take \(\mathbb{N} \subset \mathbb{R}\) using the standard metric. Find an open cover of \(\mathbb{N}\) such that the conclusion of the Lebesgue covering lemma does not hold.

Let \((X, d)\) be a metric space and \(\left\\{x_{n}\right\\}\) a sequence in \(X .\) Prove that \(\left\\{x_{n}\right\\}\) converges to \(p \in X\) if and only if every subsequence of \(\left\\{x_{n}\right\\}\) has a subsequence that converges to \(p .\)

Let \((X, d)\) be a metric space, \(S \subset X,\) and \(p \in X .\) Prove that \(p\) is a cluster point of \(S\) if and only if \(p \in \overline{S \backslash\\{p\\}}\).

Consider \(\ell^{2}\) the set of sequences \(\left\\{x_{n}\right\\}\) of real numbers such that \(\sum_{n=1}^{\infty} x_{n}^{2}<\infty\). a) Prove the Cauchy-Schwarz inequality for two sequences \(\left\\{x_{n}\right\\}\) and \(\left\\{y_{n}\right\\}\) in \(\ell^{2}\) : $$\left(\sum_{n=1}^{\infty} x_{n} y_{n}\right)^{2} \leq\left(\sum_{n=1}^{\infty} x_{n}^{2}\right)\left(\sum_{n=1}^{\infty} y_{n}^{2}\right)$$ b) Prove that \(\ell^{2}\) is a metric space with the metric \(d(x, y):=\sqrt{\sum_{n=1}^{\infty}\left(x_{n}-y_{n}\right)^{2}}\).

Let \((X, d)\) be a metric space and \(A \subset X .\) Show that \(A^{\circ}=\bigcup\\{V: V \subset A\) is open \(\\} .\)

Prove the general Bolzano-Weierstrass theorem: Any bounded sequence \(\left\\{x_{k}\right\\}\) in \(\mathbb{R}^{n}\) has a convergent subsequence.

Let \((X, d)\) be a metric space. Show that there exists a bounded metric \(d^{\prime}\) such that \(\left(X, d^{\prime}\right)\) has the same open sets, that is, the topology is the same.

Prove Proposition 7.4.6. That is, let \((X, d)\) be a complete metric space and \(E \subset X\) a closed set. Show that \(E\) with the subspace metric is a complete metric space.

Let \(\left(X, d_{X}\right)\) and \(\left(Y, d_{y}\right)\) be metric spaces, \(S \subset X, p \in X\) a cluster point of \(S,\) and let \(f: S \rightarrow Y\) be a function. Prove that \(f: S \rightarrow Y\) is continuous at \(p\) if and only if $$ \lim _{x \rightarrow p} f(x)=f(p) $$

Let \((X, d)\) be a metric space. a) Prove that for every \(x \in X,\) there either exists \(a \delta>0\) such that \(B(x, \delta)=\\{x\\}\), or \(B(x, \delta)\) is infinite for every \(\delta>0\) b) Find an explicit example of \((X, d), X\) infinite, where for every \(\delta>0\) and every \(x \in X,\) the ball \(B(x, \delta)\) is finite. c) Find an explicit example of \((X, d)\) where for every \(\delta>0\) and every \(x \in X,\) the ball \(B(x, \delta)\) is countably infinite. d) Prove that if \(X\) is uncountable, then there exists an \(x \in X\) and a \(\delta>0\) such that \(B(x, \delta)\) is uncountable.

Let \((X, d)\) be an incomplete metric space. Show that there exists a closed and bounded set \(E \subset X\) that is not compact.

Define $$ f(x, y):=\left\\{\begin{array}{ll} \frac{2 x y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0) \end{array}\right. $$ a) Show that for every fixed y the function that takes \(x\) to \(f(x, y)\) is continuous and hence Riemann integrable. b) For every fixed \(x\), the function that takes y to \(f(x, y)\) is contimous. c) Show that \(f\) is not continuous at (0,0) . d) Now show that \(g(y):=\int_{0}^{1} f(x, y) d x\) is not continuous at \(y=0 .\)

For every \(x \in \mathbb{R}^{n}\) and every \(\delta>0\) define the "rectangle" \(R(x, \delta):=\left(x_{1}-\delta, x_{1}+\delta\right) \times\left(x_{2}-\right.\) \(\left.\delta, x_{2}+\delta\right) \times \cdots \times\left(x_{n}-\delta, x_{n}+\delta\right) .\) Show that these sets generate the same open sets as the balls in standard metric. That is, show that a set \(U \subset \mathbb{R}^{n}\) is open in the sense of the standard metric if and only if for every point \(x \in U,\) there exists \(a \delta>0\) such that \(R(x, \delta) \subset U\)

Let \((X, d)\) be a metric space and \(K \subset X .\) Prove that \(K\) is compact as a subset of \((X, d)\) if and only if \(K\) is compact as a subset of itself with the subspace metric.

Let \((X, d)\) be a complete metric space. We say a set \(S \subset X\) is relatively compact if the closure \(\bar{S}\) is compact. Prove that \(S \subset X\) is relatively compact if and only if given any sequence \(\left\\{x_{n}\right\\}\) in \(S,\) there exists a subsequence \(\left\\{x_{n_{k}}\right\\}\) that converges (in \(X\) ).