Weakly bounded set
= Weakly bounded set
A subset $D$ of a <normed vector space> is weakly bounded when $\sup_{x\in D}|f(x)|<\infty$ for every $f\in X^*$. It is norm bounded: its <canonical embedding into the bidual> is a pointwise bounded family of functionals on the complete <continuous dual space>, so the <Uniform boundedness principle> applies. The converse is immediate. A <weakly compact set> is weakly bounded because each scalar evaluation has compact image.