I don't get the first one at all.

as defined is not even a vector space (the sequences

are in the space but

is not) let alone a Hilbert space for us to talk about an orthogonal subspace. Another thing, in a Hilbert space if a subspace is closed, then we can decompose the space in the direct sum of said subspace and it's orthogonal which gives that the only possible closed subspace with trivial orthogonal is the whole space.

For the second, the difficult part follows from this

Riesz's lemma - Wikipedia, the free encyclopedia
For the third one consider the canonical embedding of a space in it's double dual which gives an isometry and call

the image of

under this mapping then:

Now just use the Riesz representation theorem.

Edit:

is, of course, in the dual space.