# Problem 5, Chapter 3. Rudin’s functional analysis, point (a) and (b)

functional-analysis

Need an hint on this exercise, only first two points for now.

For $$0 < p < \infty$$, let $$l^p$$ be the sapce of all functions $$x$$ (real or complex, as the case may be) on the positive integers, such that
$$\sum_{n=1}^{\infty} |x(n)|^p < \infty$$
For $$0 \leq p < \infty$$. Define $$\left\lVert x \right\rVert = \left\{ \sum |x(n)|^p \right\}^{1/p}$$, and define $$\left\lVert x \right\rVert_{\infty} = \sup_n |x(n)|$$.
(a) Assume $$1 \leq p < \infty$$. Prove that $$\left\lVert x \right\rVert_p$$ and $$\left\rVert x \right\rVert_{\infty}$$ make $$l^p$$ and $$l^{\infty}$$ into Banach Spaces. If $$p^{-1} + q^{-1} = 1$$, prove that $$(l^p)^* = l^q$$ in the following sense: There is a one-to-one correspondence $$\Lambda \leftrightarrow y$$ between $$(l^p)^*$$ and $$l^q$$, given by
$$\Lambda x = \sum x(n)y(n) \;\;\; (x \in l^p)$$
(b) Assume $$1 < p < \infty$$ and prove that $$l^p$$ contains sequences that converge weakly but not strongly.

I think (a) is trivial. Since $$L^p(X,\mu)$$ is a banach space taking as $$\mu$$ the counting measure we obtain the result. Likewise Riesz representation with the counting measure $$\mu$$ gives me the representation of the dual space.
For (b) I am not sure, my thought was to construct an example of sequence
$$x_m(n) = \begin{cases} 1 & m \leq n \\ 0 & \text{otherwise} \end{cases}$$

I ended up with nothing, because I don't know how to show eventually this converges weakly.
(I know that this is probably a standard exercise, my apologies if the question is trivial).

#### Best Answer

Take $$x_m(n)=1$$ if $$n=m$$ and $$0$$ otherwise. This sequence converges weakly to $$0$$ becasue $$\sum x_m(n)y_n =y_m \to 0$$ as $$m \to \infty$$ for any $$y=(y_n)$$ in the dual of $$\ell^{p}$$ (which is $$\ell^{q})$$. Of course, this sequence does not converge strongly.