Continuity of a positive linear map into a dual space
= Continuity of a positive linear map into a dual space
Let $V$ be a real Banach space and $T:V\to V^*$ be linear with $T_v(v)\geq0$ for every $v$. If $v_n\to0$ and $T_{v_n}\to f$ in $V^*$, positivity of $T_{v_n+tw}(v_n+tw)$ and passage to the limit give
$$
0\leq t f(w)+t^2T_w(w)
$$
for every real $t$ and every $w$. Taking small $t$ of either sign forces $f(w)=0$. Thus the graph of $T$ is closed, and the <closed graph theorem> makes $T$ continuous.