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Liviu Nicolaescu
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This is a question of functional analysis: Start with a locally convex space $X$, form its dual $X^*$ consisting of continuous linear functionals on $X$. Now $X^*$ has several locally convex topologies $\tau$ with the following property:

For any $x\in X$ the linear map $L_x:X^*\to\mathbb{R}$ is $\tau$-continuous.

For such topology $\tau$ we form the bidual $X^{**}_\tau$ consisting of linear functionals on $X^*$ that are $\tau$ continuous. From the choice of $\tau$ we deduce that we have a natural map

$$X\ni x\mapsto L_x\in X^{**}_\tau. $$

The question is if there exist topologies $\tau$ on $X^*$ such that the above map is an isomorphism of locally convex spaces.

A natural topology on $X^*$ is the smallest locally convex topology such that all the linear functionals $L_x$ are continuous. Let's call this $\tau_0$. (Traditionally it is denoted by $\sigma(X^*, X)$.)

We can ask a more refined question: for which locally convex spaces the bidual $X^{**}_{\tau_0}$ coincides with $X$.

There is know a large class of such spaces namely the nuclear spaces. For details see

Gelfand & Shilov: Generalized Functions., vol 2.

For example the space $X=C^\infty(M)$, $M$ compact has this property.

Liviu Nicolaescu
  • 34.7k
  • 2
  • 91
  • 165