Hilbert space and its dual

[homology]

So I get the fact that kets live in a Hilbert space. Bras, which are continuous bounded linear functions of kets, must live in the <i>continuous</i> dual space. This makes sense, since we can then apply the Riesz representation theorem to Bras so that any Bra can be realized as an inner product with one entry fixed.

Now here\'s my question. The other operators, say position and momentum and such, is it safe to say that they live in the complement of the continuous dual space, i.e. they are members of the algebraic dual space of the Hilbert space, but not of the continuous dual?

If you can answer this, could you also recommend a text on quantum which is concerned with such mathematics from a mathematicians point of view?

Thanks,

Kevin