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I am reading Brezis's book and I have trouble understanding a proof. I put up an image of the relevant part of the text.

What I don't understand is towards the end when he writes:

"It follows from (4) that $\varphi$ is continuous at $0$ for the topology $\sigma(E^*,E)$..."

I understand why $W$ is a neighborhood of $0$ but I don't understand how the continuity of $\varphi$ follows from (4).

enter image description here

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up vote 1 down vote accepted

On any topological vector space (in this case $(E^*,\sigma(E^*,E)),$ any linear functional which is bounded on a neighborhood of zero is continuous. Proof: Let $\varphi$ be bounded on $U$ by $M>0.$ Then for every $\varepsilon >0$ we get $\varphi(\frac{\varepsilon}{M}U)\subset [-\varepsilon,\varepsilon],$ hence $\varphi$ is continuous at zero, hence it is continuous.

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