Problem 3
Verify that the spaces \(C_{[a, b]}, l_{2}, \mathrm{c}, c_{0}, m\) and \(R^{\infty}\) are all infinite-dimensional.
Problem 4
Given a linear space \(\mathrm{L}\), a set \(\\{\mathrm{x}\), ) of linearly independent elements of \(\mathrm{L}\) is said to be a Hamel basis (in \(\mathrm{L}\) ) if the linear subspace generated by \(\\{\mathrm{x}\), ) coincides with L. Prove that a) Every linear space has a Hamel basis; b) If \(\\{x\), ) is a Hamel basis in \(\mathrm{L}\), then every vector \(\mathrm{x} E \mathrm{~L}\) has a unique representation as a finite linear combination of vectors from the set \(\left\\{x_{\alpha}\right\\}\) c) Any two Hamel bases in a linear space \(\mathrm{L}\) have the same power (cardinal number), called the algebraic dimension of \(\mathrm{L}\); d) Two linear spaces are isomorphic if and only if, they have the same algebraic dimension.
Problem 5
Let \(\boldsymbol{L}^{\prime}\) be a \(k\)-dimensional subspace of an \(\mathrm{n}\)-dimensional linear space L. Prove that the factor space \(L / L^{\prime}\) has dimension \(n-\mathrm{k}\).
