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Let $X$ be a locally convex space. Let $M$ be a dense subspace of $X$ and let $f\in M^*$. And I am trying to show there exists $g\in X^*$ such that $g|_M=f$.

My attempts are:

Let $x\in X$. Then there exists a net $(x_i)$ in $M$ such that $x_i\rightarrow x$. Thus $(x_i)$ is a Cauchy net in $M$, and hence $(f(x_i))$ is a Cauchy net in $\mathbb{F}$. Since $\mathbb{F}$ is a Banach space, $\lim\limits_i f(x_i)$ exists in $\mathbb{F}$. Now define $g(x)=\lim\limits_i f(x_i)$. Then $g:X\rightarrow\mathbb{F}$ is linear and $g|_M=f$. Let $(x_k)$ be a net in $X$ such that $x_k\rightarrow x$.

And I am trying to show $g(x_k)\rightarrow g(x)$ to get continuity of $g$. But I am stuck here. Does $g(x_k)\rightarrow g(x)$? If so how to prove it?

Thank you in advance!

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  • $\begingroup$ Why is $g$ well-defined and linear? I cannot see that. $\endgroup$ – amsmath Sep 11 '18 at 4:30

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