Let \(A\) be a set of \(a<\infty\) elements and \(B\) a set of \(b<\infty\) elements. Prove that there are exactly \(b^a\) distinct maps from \(A\) to \(B\text{.}\)
2.Topology of a metric space.
Let \((M,d)\) be a metric space and let \(\cU\) be the collection of all sets \(U\) such that, for every \(x\in U\text{,}\) there is an \(\eps>0\) such that \(B_\eps(x)\subset U\text{.}\) Prove that \(\cU\) is a topology on \(M\text{.}\)
3.Continuity of the distance function.
Let \((M,d)\) be a metric space. Prove that the distance function \(d:M\times M\to[0,\infty)\) is continuous with respect to the induced topology.
4.Topology fron norm.
Let \((V,\|\cdot\|)\) be a normed space. Prove that the induced topology makes \(V\) a topological vector space, namely prove that the sum of vectors and the multiplication by a scalar are continuous.
5.Norm from scalar product.
Let \((V,\langle\cdot\rangle)\) be a vector space with a scalar product. Prove that \(\|v\| = \langle v,v \rangle^{1/2}\) is a norm on \(V\text{.}\)
