# If every linear functional is continuous in $V$, is every linear functional continuous for $S\subseteq V$?

Suppose $V$ is a finite dimensional (real) topological vector space. The first lemma in these notes says that

Every vector subspace of a tvs with the induced topology is a topological vector space in its own right.

But no proof is offered. With Michael Greinecker's answer, it makes sense to me why any subspace of a tvs is a tvs.

I want to further ask, if the topology on $V$ is such that every linear functional is continuous, does the induced topology on $S$ also have that all linear functionals on $S$ are continuous? I think the answer is yes. If $f_S$ is a linear functional on $S$, then $f_S$ can be extended to a linear functional $f$ on $V$. Then for $O\cap S$ open in $S$, $$f_S^{-1}(O\cap S)=f^{-1}(O)\cap S$$ which is open in $S$? Have I made sense or nonsense? If it is nonsense, could someone give a brief proof so I can see how it is done properly?

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