Hahn–Banach Operators: A Review

ð2Þ ~ T 1⁄4 T ; 8 2 V: We use HBðV;W Þ to denote the set of Hahn–Banach operators. The classical Hahn–Banach theorem states that every rank-1 operator from V into W is a Hahn–Banach operator. It can also be restated in the following way: if dimW 1⁄4 1, then HBðV;W Þ 1⁄4 LðV;W Þ. Observe that the two statements above are slightly different. In the first case we describe a property of an operator T (being of rank 1) and conclude that, for every pair V and W, such an operator is a Hahn–Banach operator. The second statement starts with the description of the Banach space W (being one-dimensional) and concludes that for every Banach space V and every operator T 2 LðV;W Þ, T is a Hahn–Banach operator.