Problem 1
Prove that in a Euclidean space, the operations of addition, multiplication by numbers and the formation of scalar products are all continuous. More exactly, prove that if \(\boldsymbol{x}, \rightarrow \boldsymbol{x}, \boldsymbol{y}, \rightarrow \boldsymbol{y}\) (in the sense of norm convergence) and \(A, \rightarrow A\) (in the sense of ordinary convergence), then $$ x_{n}+y_{n} \rightarrow x+y, \quad \lambda_{n} x_{n} \rightarrow \lambda x, \quad\left(x_{n}, y_{n}\right) \rightarrow(x, y) $$ Hint. Use Schwarz's inequality.
Problem 2
Let \(\boldsymbol{R}\) be the set of all functions \(\mathbf{f}\) defined on the interval \([0,1]\) such that 1) \(\mathbf{f}(t)\) is nonzero at no more than countably many points \(t_{1}, t_{2}, \ldots\); 2) \(\sum_{i=1}^{\infty} f^{2}\left(t_{i}\right)<\infty .\) Define addition of elements and multiplication of elements by scalars in the ordinary way, i.e., \((f+g)(t)=f(t)+g(t),(a f)(t)=a f(t)\). Iff and \(g\) are two elements of \(\boldsymbol{R}\), nonzero only at the points \(t_{1}, t_{2}, \ldots\) and \(t_{1}^{\prime}, t_{2}^{\prime}, \ldots\) respectively, define the scalar product off and \(g\) as $$ (f, g)=\sum_{i, j=1}^{\infty} f\left(t_{i}\right) g\left(t_{j}^{\prime}\right) $$ Prove that this scalar product makes \(R\) into a Euclidean space. Prove that \(R\) is nonseparable, i.e., that \(\mathrm{R}\) contains no countable everywhere dense subset.
Problem 5
Given a Euclidean space \(R\), let \(\varphi_{1}, \varphi_{2}, \ldots, \varphi_{k}, \ldots\) be an orthonormal basis in \(R\) and \(\mathbf{f}\) an arbitrary element of \(R\). Prove that the element $$ f-\sum_{k=1}^{n} a_{k} \varphi_{k} $$ is orthogonal to all linear combinations of the form if and only if $$ \sum_{\mathrm{k}=1} b_{k} \varphi_{k} $$ $$ a_{k}=\left(f, \varphi_{k}\right) \quad(k=1,2, \ldots, n) $$
Problem 8
Give an example of a Euclidean space R and an orthonormal system \(\left\\{\varphi_{n}\right\\}\) in \(R\) such that \(R\) contains no nonzero element orthogonal to every \(\varphi_{n}\), even though \(\left\\{\varphi_{n}\right\\}\) fails to be complete.
Problem 10
Prove that each of the following sets is a subspace of the Hilbert space \(l_{2}\) : a) The set of all \(\left(x_{1}, x_{2}, \ldots, x_{k}, \ldots\right) \in \mathrm{I}\), such that \(x_{1}=x_{2}\); b) The set of all \(\left(\mathrm{x},, x_{2}, \ldots, x_{k}, \ldots\right) \in l_{2}\) such that \(x_{k}=0\) for all even \(\mathrm{k}\).
