Problem 4
Let \(R\) be a Banach space, and let \(M\) be a closed subspace of \(R\). Define a norm in the factor space \(P=R / M\) by setting $$ \|\xi\|=\inf _{x \in \xi}\|x\| $$ for every element (residue class) \(\xi \in P\). Prove that a) \(\|\xi\|\) is actually a norm in \(P\); b) The space \(P\), equipped with this norm, is a Banach space.
Problem 5
Let \(R\) be a normed linear space. Prove that a) Every finite-dimensional linear subspace of \(R\) is closed; b) If \(M\) is a closed subspace of \(R\) and \(N\) a finite-dimensional subspace of \(R\), then the set $$ M+N=\\{z: z=x+y, x \in M, y \in N\\} $$ is a closed subspace of \(R\); c) If \(\mathrm{Q}\) is an open convex set in \(\mathrm{R}\) and \(x_{0} \notin \mathrm{Q}\), then there exists a closed hyperplane which passes through the point \(x_{0}\) and does not intersect \(Q .\)
Problem 7
Two norms \(\|\cdot\|_{1},\|\cdot\|_{2}\) in a linear space \(\mathrm{R}\) are said to be equivalent if there exist constants \(a, b>0\) such that $$ a\|x\|_{1}<\|x\|_{2} \leqslant b\|x\|_{1} $$ for all \(x \in R\). Prove that if \(R\) is finite-dimensional, then any two norms in \(R\) are equivalent.
