Hello!

We have the following space:

,

considered as a subspace of the Hilbert space (the space of all sequences such that , with the norm ), where .

I have to find its orthogonal complement, and decide whether the space is dense and closed in .

Now, here's what I got:

and

i.e. they are strictly monotone decreasing and positive.

Hence, their linear combinations would again be either strictly decreasing or strictly increasing, and would either be all positive (in the former case) or all negative (in the latter).

Hence, we cannot approximate, say, , by linear combinations of sequences in Z. Therefore, we have that Z is not dense in .

From this (and a lemma, which says that a space is dense iff its orthogonal complement is {0}) we can conclude that the orthogonal complement is not .

But how do I go about finding the orthogonal complement? We haven't covered orthonormal bases of Hilbert spaces yet... My intuition is the following:

But how do I show all this rigorously??? Does the reverse inclusion hold, as well?

And I have no intuition about whether it's closed or not...