Let $X$ and $Y$ be normed spaces. What is an isomorphism between $X$ and $Y$ ? Show that a bounded linear map $T: X \rightarrow Y$ is an isomorphism if and only if $T$ is surjective and there is a constant $c>0$ such that $\|T x\| \geqslant c\|x\|$ for all $x \in X$. Show that if there is an isomorphism $T: X \rightarrow Y$ and $X$ is complete, then $Y$ is complete.

Show that two normed spaces of the same finite dimension are isomorphic. [You may assume without proof that any two norms on a finite-dimensional space are equivalent.] Briefly explain why this implies that every finite-dimensional space is complete, and every closed and bounded subset of a finite-dimensional space is compact.

Let $Z$ and $F$ be subspaces of a normed space $X$ with $Z \cap F=\{0\}$. Assume that $Z$ is closed in $X$ and $F$ is finite-dimensional. Prove that $Z+F$ is closed in $X$. [Hint: First show that the function $x \mapsto d(x, Z)=\inf \{\|x-z\|: z \in Z\}$ restricted to the unit sphere of F achieves its minimum.]