91Ó°ÊÓ

(a) Let \(F\) be the field of all real numbers and let \(V\) be the set of all sequences \(\left(a_{1}, a_{2}, \ldots, a_{n}, \ldots\right), a, \in F\), where equality, addition and scalar multiplication are defined componentwise. Prove that \(V\) is a vector space over \(F\). (b) Let \(W=\left\\{\left(a_{1}, \ldots, a_{n}, \ldots\right) \in V \mid \lim _{n \rightarrow \infty} a_{n}=0\right\\}\). Prove that \(W\) is a subspace of \(V\). *(c) Let \(U=\left\\{\left(a_{1}, \ldots, a_{n, \ldots}\right) \in V \mid \sum_{i=1}^{\infty} a_{i}^{2}\right.\) is finite\\}. Prove that \(U\) is a subspace of \(V\) and is contained in \(W\).

If \(V\) is finite-dimensional and \(v_{1} \neq v_{2}\) are in \(V\) prove that there is \(\operatorname{an} f \in \hat{V}\) such that \(f\left(v_{1}\right) \neq f\left(v_{2}\right)\).

Let \(M, N, Q\) be three \(R\) -modules, and let \(T\) be a homomorphism of \(M\) into \(N\) and \(S\) a homomorphism of \(N\) into \(Q .\) Define \(T S: M \rightarrow Q\) by \(m(T S)=(m T) S\) for any \(m \in M .\) Prove that \(T S\) is an \(R\) -homomorphism of \(M\) into \(Q\) and determine its kernel, \(K(T S)\).

If \(V\) is of dimension \(n\) show that any set of \(n\) linearly independent vectors in \(V\) forms a basis of \(V\),