Before we proceed with the question, let us introduce the following notations:
Notations.
- $X$ and $Y$ are TVS over the field $\mathbb{R}$.
- $L(X,Y)$ is the space of all linear maps from $X$ into $Y$.
- $CL(X,Y)$ is the space of all continuous linear maps from $X$ into $Y$.
- $X^*= L(X,\mathbb{R})$, called the algebraic dual of $X.$
- $X'=CL(X,\mathbb{R})$, called the topological dual of $X.$
For a seminorm $p$ on $X$, we put $$U_p=\{x\in X: p(x)\le 1\}$$ and $$ U_{p}^o=\{x^*\in X^*:|x^*(x)|\le 1\ \ \forall x\in U_p\}.$$
My question is about a statement stated in the book of Kluvanek and Knowles, entitled "Vector Measures and Control System." It is stated as follows.
Statement. Let $(X,\tau)$ be a locally convex topological vector space (LCTVS) and let $W'\subseteq X'$. Then $W'$ is equicontinuous iff there exists a continuous seminorm $p$ on $X$ such that $W'\subseteq U_{p}^o.$
Can you please provide me an explanation why the preceding statement is true. Thanks in advance.
Added. I am using the definition of Schwartz in his Functional Analysis book.
Let $X$ and $Y$ be TVS. A subset $\mathcal{F}\subseteq CL(X,Y)$ is said to be equicontinuous if for every neighborhood $V$ of $0$ in $Y$, there is a neighborhood $U$ of $0$ in $X$ such that $f(U)\subseteq V$ for all $f\in \mathcal{F}.$
Attention please... As of this time, I cannot post any comments below nor to accept answers. What's wrong? Earlier, I cannot even properly tag this post. I don't know how to fix it...Sorry for the inconvenience
