Let and be normed spaces. Assigning to each continuous linear operator the scalar
defines a norm on that makes into a normed space. Moreover, if is a Banach space then so is (en)
Let be a normed space and for every let
where by definition is a scalar.
Then
is a norm that makes a Banach space.
If is the closed unit ball of then for every
Consequently, is a bounded linear functional on with norm
is weak*-compact. (en)
