91Ó°ÊÓ

Determine the null space of the operator \(T: \mathbf{R}^{3} \longrightarrow \mathbf{R}^{2}\) represented by $$ \left[\begin{array}{rrr} 1 & 3 & 2 \\ -2 & 1 & 0 \end{array}\right] $$

Give examples of compact and noncompact curves in the plane \(\mathbf{R}^{2}\).

Show that equivalent norms on a vector space \(X\) induce the same topology for \(X\).

Let \(X\) and \(Y\) be normed spaces and \(T_{n}: X \longrightarrow Y(n=1,2, \cdots)\) bounded linear operators. Show that convergence \(T_{n} \longrightarrow T\) implies that for every \(\varepsilon>0\) there is an \(N\) such that for all \(n>N\) and all \(x\) in any given closed ball we have \(\left\|T_{n} x-T x\right\|<\varepsilon\).

Let \(Z\) be a proper subspace of an \(n\)-dimensional vector space \(X\), and let \(x_{0} \in X-Z\). Show that there is a linear functional \(f\) on \(X\) such that \(f\left(x_{0}\right)=1\) and \(f(x)=0\) for all \(x \in Z\).

Let \(X\) be the vector space of all complex \(2 \times 2\) matrices and define \(T: X \longrightarrow X\) by \(T x=b x\), where \(b \in X\) is fixed and \(b x\) denotes the usual product of matrices. Show that \(T\) is linear. Under what condition does \(T^{-1}\) exist?

Show that two linear functionals \(f_{1} \neq 0\) and \(f_{2} \neq 0\) which are defined on the same vector space and have the same null space are proportional.

(Half space) Let \(f \neq 0\) be a bounded linear functional on a real normed space \(X\). Then for any scalar \(c\) we have a hyperplane \(H_{e}=\\{x \in X \mid f(x)=c\\}\), and \(H_{c}\) determines the two half spaces $$ X_{c 1}=\\{x \mid f(x) \leqq c\\} \quad \text { and } \quad X_{c 2}=\\{x \mid f(x) \geqq c\\} $$ Show that the closed unit ball lies in \(X_{e 1}\) where \(c=\|f\|\), but for no \(\varepsilon>0\), the half space \(X_{e 1}\) with \(c=\|f\|-\varepsilon\) contains that ball.