Rigged Hilbert Space: Algebraic v.s. Continuous Dual Space

[Rasalhague]

Definitions of a rigged Hilbert space typically talk about the "dual space" of a certain dense subspace of a given Hilbert space H. Do they mean the algebraic or the continuous dual space (continuous wrt the norm topology on H)?

 

[micromass]

Since they identify $H^*$ with H (by the Riesz representation theorem), they are talking about the continuous dual. That is: the bounded linear functionals from H to $\mathbb{C}$.

 

[Rasalhague]

Is it continuous with respect to the metric topology induced by the norm from the inner product on H, or continuous with respect to the other topology for the dense subset mentioned in the Wikipedia article?

 

[micromass]

They say in the article that it is the continuous dual wrt the finer topology. This would make sense.

 

[dextercioby]

Look on the internet for the PhD thesis of Prof. Rafael de la Madrid. Though some small corrections can be made to it, it's the gem for RHS.

 

[Rasalhague]

In equations (4.2) and (4.3), de la Madrid defines a notion of convergence for sequences of elements of the subspace $\Phi$, which he says induces a topology, T, for $\Phi$. How does convergence induce a topology? I guess one works backwards somehow from the definition of sequence convergence: a sequence $(s_n )$ converges to $x\in X$ iff $(s_n )$ is in every neighborhood residually.

Is $||\cdot ||_{l,m,n}$ a norm for every $l,m,n \in \mathbb{N}$, as the notation suggests? If so, what is the relationship of T to the metric topologies induced by these norms? (Do they induce different topologies? Is T perhaps the intersection of these topologies?)

EDIT: Ah, reading Wikipedia: Sequential space and thence the first Franklin article, Spaces in which sequences suffice, specifically condition (b) in section 0, could it be that de la Madrid's topology comes from defining an open set as one for which every sequence converging to a point in the set is eventually/residually in the set?

The definition of a http://planetmath.org/FrechetSpace.html looks interesting too, in particular the determination of a topology by a countable family of seminorms. I wonder if the topology de la Madrid refers to is determined in such a way. Are his maps $||\cdot ||_{l,m,n}$ at least seminorms?