91Ó°ÊÓ

Suppose \(V\) is a normed vector space and \(\varphi\) is a linear functional on \(V\). Suppose \(\alpha \in \mathbf{F} \backslash\\{0\\}\). Prove that the following are equivalent: (a) \(\varphi\) is a bounded linear functional. (b) \(\varphi^{-1}(\alpha)\) is a closed subset of \(V\). (c) \(\overline{\varphi^{-1}(\alpha)} \neq V\).

Show that if \(a, b \in \mathbf{R}\) with \(a+b i \neq 0,\) then $$ \frac{1}{a+b i}=\frac{a}{a^{2}+b^{2}}-\frac{b}{a^{2}+b^{2}} i $$

Show that the map \(f \mapsto\|f\|\) from a normed vector space \(V\) to \(\mathbf{F}\) is continuous (where the norm on \(\mathbf{F}\) is the usual absolute value).

Suppose \(U\) is a subset of a metric space \(V\). Show that \(U\) is dense in \(V\) if and only if every nonempty open subset of \(V\) contains at least one element of \(U\).

Suppose \(z \in \mathbf{C}\). Prove that $$ \max \\{|\operatorname{Re} z|,|\operatorname{Im} z|\\} \leq|z| \leq \sqrt{2} \max \\{|\operatorname{Re} z|,|\operatorname{Im} z|\\} $$.

Suppose \(\varphi\) is a linear functional on a vector space \(V .\) Prove that if \(U\) is a subspace of \(V\) such that null \(\varphi \subset U,\) then \(U=\) null \(\varphi\) or \(U=V\) .

Suppose \(U\) is a subset of a metric space \(V\). Show that \(U\) has an empty interior if and only if \(V \backslash U\) is dense in \(V\).

Prove that every finite subset of a metric space is closed.

Prove or give a counterexample: If \(V\) is a metric space and \(U, W\) are subsets of \(V,\) then \((\operatorname{int} U) \cup(\operatorname{int} W)=\operatorname{int}(U \cup W)\).

Prove that every closed ball in a metric space is closed.