Chapter 1

1-23. If \(f: A \rightarrow \mathbf{R}^{m}\) and \(a \in A\), show that \(\lim _{x \rightarrow a} f(x)=b\) if and only if \(\lim _{x \rightarrow a} f^{i}(x)=b^{i}\) for \(i=1, \ldots, m\).

1-26. Let \(A=\left\\{(x, y) \in \mathbf{R}^{2}: x>0\right.\) and \(\left.0

1-17. Construct a set \(A \subset[0,1] \times[0,1]\) such that \(A\) contains at most one point on each horizontal and each vertical line but boundary \(A=[0,1] \times[0,1] .\) Hint: It suffices to ensure that \(A\) contains points in each quarter of the square \([0,1] \times[0,1]\) and also in each sixteenth, etc.

Prove that ||\(x|-| y|| \leq|x-y|\).

1-27. Prove that \(\left\\{x \in \mathbf{R}^{n}:|x-a|

1-17. Construct a set \(A \subset[0,1] \times[0,1]\) such that \(A\) contains at most one point on each horizontal and each vertical line but boundary \(A=[0,1] \times[0,1] .\) Hint: It suffices to ensure that \(A\) contains points in each quarter of the square \([0,1] \times[0,1]\) and also in each sixteenth, etc.

1-28. If \(A \subset \mathbf{R}^{n}\) is not closed, show that there is a continuous function \(f: A \rightarrow \mathbf{R}\) which is unbounded. Hint: If \(x \in \mathbf{R}^{n}-A\) but \(x \notin\) interior \(\left(\mathbf{R}^{n}-A\right)\), let \(f(y)=1 /|y-x|\).

1-18. If \(A \subset[0,1]\) is the union of open intcrvals \(\left(a_{i}, b_{i}\right)\) such that each rational number in \((0,1)\) is contained in some \(\left(a_{i}, b_{i}\right)\), show that boundary \(A=[0,1]-A\).

Let \(f\) and \(g\) be integrable on \([a, b]\) (a) Prove that \(\left|\int_{a}^{b} f \cdot g\right| \leq\left(\int_{a}^{b} f^{2}\right)^{\frac{1}{2}} \cdot\left(\int_{a}^{b} g^{2}\right)^{\frac{1}{2}} .\) Hint: Consider separately the cases \(0=\int_{a}^{b}(f-\lambda g)^{2}\) for some \(\lambda \in \mathbf{R}\) and \(0<\) \(\int_{a}^{b}(f-\lambda g)^{2}\) for all \(\lambda \in \mathbf{R}\). (b) If equality holds, must \(f=\lambda g\) for some \(\lambda \in \mathbf{R} ?\) What if \(f\) and \(g\) are continuous?

1-19.* If \(A\) is a closed set that contains every rational number \(r \in[0,1]\), show that \([0,1] \subset A\)

A linear transformation \(T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{n}\) is norm preserving if \(|T(x)|=|x|\), and inner product preserving if \(\langle T x, T y\rangle=\langle x, y\rangle\). (a) Prove that \(T\) is norm preserving if and only if \(T\) is innerproduct preserving. (b) Prove that such a linear transformation \(T\) is \(1-1\) and \(T^{-1}\) is of the same sort.

If \(x, y \in \mathbf{R}^{n}\) are non-zero, the angle between \(x\) and \(y\), denoted \(\angle(x, y)\), is defined as arccos \((\langle x, y\rangle /|x| \cdot|y|)\), which makes sense by Theorem 1-1(2). The linear transformation \(T\) is angle preserving if \(T\) is 1-1, and for \(x, y \neq 0\) we have \(\angle(T x, T y)-\angle(x, y)\). (a) Prove that if \(T\) is norm preserving, then \(T\) is angle preserving. (b) If there is a basis \(x_{1}, \ldots, x_{n}\) of \(\mathbf{R}^{n}\) and numbers \(\lambda_{1}, \ldots, \lambda_{n}\) such that \(T x_{i}=\lambda_{i} x_{i}\), prove that \(T\) is angle preserving if and only if all \(\left|\lambda_{i}\right|\) are equal. (c) What are all angle preserving \(T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{n} ?\)

1-21. \(^{*}\) (a) If \(A\) is closed and \(x \notin A\), prove that there is a number \(d>0\) such that \(|y-x| \geq d\) for all \(y \in A\). (b) If \(A\) is closed, \(B\) is compact, and \(A \cap B=\varnothing\), prove that there is \(d>0\) such that \(|y-x| \geq d\) for all \(y \in A\) and \(x \in B\) Hint: For each \(b \in B\) find an open set \(U\) containing \(b\) such that this relation holds for \(x \in U \cap B\). (c) Give a counterexample in \(\mathbf{R}^{2}\) if \(A\) and \(B\) are closed but neither is compact.

1-22. \({ }^{*}\) If \(U\) is open and \(C \subset U\) is compact, show that there is a compact set \(D\) such that \(C \subset\) interior \(D\) and \(D \subset U\).

. *\( If \)T: \mathbf{R}^{m} \rightarrow \mathbf{R}^{n}\( is a linear transformation, show that there is a number \)M\( such that \)|T(h)| \leq M|h|\( for \)h \in \mathbf{R}^{m} .\( Hint \):\( Estimate \)|T(h)|\( in terms of \)|h|\( and the entries in the matrix of \)T$.

* If \(x, y \in \mathbf{R}^{n}\), then \(x\) and \(y\) are called perpendicular (or orthogonal) if \(\langle x, y\rangle=0 .\) If \(x\) and \(y\) are perpendicular, prove that \(|x+y|^{2}=|x|^{2}+|y|^{2}\)