PROBLEM 5.8. Write out a proof (you can steal it from one of many places but at least write it out in your own hand) either for $p=2$ or for each $p$ with $1 \leq p<\infty$ that
$$l^p=\left{a: \mathbb{N} \longrightarrow \mathbb{C} ; \sum_{j=1}^{\infty}\left|a_j\right|^p<\infty, a_j=a(j)\right}$$
is a normed space with the norm
$$|a|_p=\left(\sum_{j=1}^{\infty}\left|a_j\right|^p\right)^{\frac{1}{p}}$$
This means writing out the proof that this is a linear space and that the three conditions required of a norm hold.

Problem 5.12. Consider the ‘unit sphere’ in $l^p$. This is the set of vectors of length 1 :
$$S=\left{a \in l^p ;|a|_p=1\right}$$
(1) Show that $S$ is closed.
(2) Recall the sequential (so not the open covering definition) characterization of compactness of a set in a metric space (e.g. by checking in Rudin’s book).
(3) Show that $S$ is not compact by considering the sequence in $l^p$ with $k$ th element the sequence which is all zeros except for a 1 in the $k$ th slot. Note that the main problem is not to get yourself confused about sequences of sequences!

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