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There is a guided exercise in the course of functional analysis that I am following where we have to prove that a topological vectors space is seminormed if and only if it is locally convex. There is one step in te proof that I can't figure out.

Let [itex]X[/itex] be a topological vector space and [itex]\mathcal{U}[/itex] a convex neighbourhood of [itex]0[/itex].

Define

[tex]\mathcal{V} = \bigcap_{\lambda \in S_1} \lambda \mathcal{U} [/tex].

Where [itex]S_1[/itex] are the complex numbers of modulus one.

Use the compactness of [itex]S_1[/itex] to prove that [itex]\mathcal{V}[/itex] is a balanced convex neighbourhood of zero.

Proving that [itex]\mathcal{V}[/itex] is balanced an convex is straightforward and doesn't require the compactness of [itex]S_1[/itex]. I haven't been able to prove that [itex]\mathcal{V}[/itex] is still a neighbourhood of zero though.

Thanks,

A_B

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# Proof that a TVS is seminormed iff it is locally convex

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