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Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$.

Let $f$ be a sublinear function $m(X,X^*)$-continuous on the subset $H$ of $X$ that goes to $\mathbb{R}$.

Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?

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    $\begingroup$ Banach spaces are Mackey, i.e., the Mackey topology is the norm topology. $\endgroup$
    – Jochen Wengenroth
    Commented May 29, 2020 at 6:17
  • $\begingroup$ Then, $f$ admits a continuous extension to $X$? $\endgroup$
    – kaka Hae
    Commented May 29, 2020 at 13:18

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