I'm trying to understand the proof of Theorem 3.5 in Rudin's Functional Analysis.
The context is: $M$ is a subspace of a locally convex space $X$, and $T$ is a continuous linear functional in the dual $X^*$. The claim that I'm having trouble seeing is:
Since $T(M)$ is a proper subspace of the scalar field, we must have $T(M) = \{0\}.$
Why is this? I know that every non-zero subspace must be unbounded... is $T(M)$ bounded for some reason that I don't see?
If $T(M)\ne\{0\}$, say, $c\in T(M)$, $c\ne 0$, say, $T(x)=c$, then for every scalar $a$, $T(ac^{-1}x)=ac^{-1}T(x)=ac^{-1}c=a$, then $a\in T(M)$ since $ac^{-1}x\in M$, so $T(M)$ is the whole scalar field, a contradiction.