Exercises 1

Exercise 1.1: Prove Theorem 1.4.

Exercise 1.2: Prove Theorem 1.5.

Exercise 1.3: Prove Theorem 1.7.

Exercise 1.4: Fill in the details of the assertions in Section 1.2.

Exercise 1.5: Fill in the details of the assertions in Example 1.20.

Exercise 1.6: Let $\phi$ be a bounded sesquilinear form in $\mathcal{H}$. Using the Riesz representation theorem, prove that there exists a unique bounded linear operator $T \in \mathcal{B}(\mathcal{H})$ such that $\phi(x,y) = (Tx,y)$ for all $x,y \in \mathcal{H}$. Prove furthermore that the constant $C$ in 1.3 is equal to $\|T\|$.

Exercise 1.7: Prove that a bounded linear operator $T$ in $\mathcal{B}(\mathcal{H})$ is uniquely determined by its quadratic form, i.e. the map $x \mapsto (Tx,x)$. Hint: derive a version of the polarization identity.

Exercise 1.8: Using the principle of uniform boundedness, prove that if $T_n \rightarrow T$ strongly, then $\sup_n \|T_n\| < \infty$. Repeat this for weak convergence.

Exercise 1.9: Prove that if $T_n \rightarrow T$ strongly then $\|T\| \leq \lim \inf \|T_n\|$. Give an example where $\|T_n\| \rightarrow \|T\|$ is false.

Exercise 1.10: Let $X$ be a Banach space and let $T_n : X \rightarrow X$ be a sequence of bounded linear operators such that for all $x \in X$, the limit $\lim_{n \rightarrow \infty} T_n x$ exists in the norm of $X$. Using the principle of uniform boundedness, prove that there exists a bounded operator $T: X \rightarrow X$ such that $T_n \rightarrow T$ strongly as $n \rightarrow \infty$.

Exercise 1.11: Show that the range of the operator $B : \ell^p \rightarrow \ell^p, 1 \leq p < \infty$,

is not closed.

Exercise 1.12:

1. Let $T_n \rightarrow T$ and $S_n \rightarrow S$ strongly. Prove that $T_nS_n \rightarrow TS$ strongly.
2. Let $T_n \rightarrow T$ and $S_n \rightarrow S$ weakly. Give an example where the weak convergence $T_nS_n \rightarrow TS$ does not hold.

Exercise 1.13: For $k \in \mathbb{N}$, let $t^{(k)} \in \ell^{\infty}$ (i.e. for each $k$, $t^{(k)}$ is a sequence $(t_1^{(k)},t_2^{(k)},...) \in \ell^{\infty}$). Let $T^{(k)}$ be the operator of multiplication by $t^{(k)}$ in $\ell^2$ (see Example 1.14). Give a necessary and sufficient condition (in terms of the squence $t^{(k)}$) for strong convergence $T^{(k)} \rightarrow 0$.

Exercise 1.14: For $k \in \mathbb{N}$, let $t^{(k)} \in L^{\infty}(\mathbb{R})$ and let $T^{(k)}$ be the operator of multiplication by $t^{(k)}$, see Example 1.15. Give a necessary and sufficient condition for uniform convergence $T^{(k)}\rightarrow 0$ in terms of $t^{(k)}$. Give a sufficient condition (in terms of the sequence $t^{(k)}$) for strong convergence $T^{(k)} \rightarrow 0$.