# Locally convex space characterization in terms of duality

Let $(X,\tau)$ be a topological vector space (TVS for short). Denote by $X^*$ the topological dual of $(X,\tau)$. If there exists a locally convex topology $\mu$ on $X$ compatible with the duality $(X,X^*)$ (that is, $(X,\mu)^*=X^*$) and $\mu$ is finer than $\tau$ then is $(X,\tau)$ a locally convex space?

An (equivalent) reformulation of the above question would be: - Is there an (infinite dimensional) locally convex space $(X,\mu)$, such that in between the weak topology, $w$ and $\mu$ there exists a linear topology $\tau$ on $X$ which is NOT locally convex?

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