How does one go about proving that a space is a dual space? The only thing I can think of is to prove that the space is isomorphic to a dual space. Is there a better way to do this?
Thank you
2026-04-11 12:55:52.1775912152
How to prove a space is a dual space?
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Well, it is enough to show that the unit ball is compact with respect to some locally convex topology, that is weaker than the norm topology. See this article.
For example, if $X$ is a reflexive space with a basis, then the unit ball of $\mathscr B(X)$, the space of bounded linear operators on $X$, furnished with the weak operator topology is compact. This means that $\mathscr B(X)$ is a dual space. (Of course in this case it is not very difficult to identify the predual explicitly.)