# Explanations for a "diagonal process" construction of a sequence

I am reading Peter Lax's "Functional analysis".

Let $y_n$ be a bounded sequence of vectors in a Banach reflexive space, $X, Y$ their closed linear span. Take a countable set $m_j$ of applications belonging to the dual $Y^*$ of $Y$ ($Y^*$ is separable).

The author then says that we can apply the "classical diagonal process" to obtain a subsequence $z_n$ of vectors such that $\lim_n m_j(z_n)$ exists for every $j$.

I do not understand how he builds this subsequence.

• How is $X$ defined ? Jul 20, 2016 at 15:16
• X is a reflexive Banach space Jul 20, 2016 at 15:29

The key ingredient is the observation that if a sequence is convergent, then any subsequence is also convergent.

1. First, you construct recursively a sequence of subsequences of $y_n$ as follows.
2. To begin, put $z_{n,0}:=y_n$.
3. Given $(z_{n,j})_n$, you can find (by compactness) a subsequence $(z_{n,j+1})_n$ such that $\lim_{n}m_j(z_{n,j+1})$ exists.
4. Note that then all $\lim_n m_{j'}(z_{n,j+1})$ exist, for $j'\leq j$ (because in this case $(z_{n,j+1})_n$ is a subsequence of $(z_{n,j'+1})_n$).
5. Finally, put $z_n:=z_{n,n}$ (i.e. $z_n$ is the diagonal of $z_{n,j}$, hence the name). To see that $\lim_n m_j(z_n)$ exists, note that except for finitely many elements, $z_n$ is a subsequence of $z_{n,j}$.

You don't need $X$ to be reflexive, or even a Banach space for this argument. All that really matters is that all the functions $m_j$ map the sequence $y_n$ into compact metric spaces (so that we can use compactness in step 3.) -- the topology on $X$ (or its linear structure) is completely irrelevant beyond this, guaranteed by the fact that bounded functionals by definition map bounded sequences to bounded subsets of the real line.

• Thank you very much Tomasz, now it is clear. It has been really instructive. Jul 20, 2016 at 18:35
• The only thing it would like to understand better is passage 4. Could you be more explicit please? Jul 20, 2016 at 19:14
• @GiovanniSiclari: There you go. Jul 21, 2016 at 9:25
• Thank you. This morning I have finally understood it. Jul 21, 2016 at 9:27