Problem 2
Let A be a linear operator mapping \(m\)-space \(\mathrm{R}^{\mathrm{m}}\) into \(\mathrm{n}\)-space \(R^{n}\). Prove that the image of \(R^{m}\), i.e., the set \(\left\\{y: y=\right.\) Ax, \(x \in R^{m}\) ), has dimension no greater than \(\mathrm{m}\).
Problem 6
Let \(C_{[a, b]}^{\\{1]}\) be the set of all continuously differentiable functions on \([\mathrm{a}, b]\), and let \(\mathrm{D}\) be the differentiation operator, defined by, \(D f(x)=f^{\prime}(x)\) for all \(f \in C_{[a, b]}^{(1)}\). Prove that a) \(C_{[a, b]}^{(1)}\) is a linear space, b) \(\mathrm{D}\) is a linear operator mapping \(C_{[a, b]}^{(1)}\) onto \(C_{[a, b]}\); c) \(\mathrm{D}\) is not continuous on \(C_{[a, b]}\) d) \(\mathrm{D}\) is continuous with respect to the norm.
Problem 8
Interpret the differentiation operator as a continuous linear operator on the space of all generalized functions. Hint. Take continuity to mean that if a sequence of generalized functions \(\left\\{f_{n}(x)\right\\}\) converges to a generalized function \(f(x)\), then \(\left\\{f_{n}^{\prime}(x)\right)\) converges to \(f^{\prime}(x)\).
