Dual Spaces

Abstract

Let X be a real (or complex) normed vector space. A bounded linear operator from X into the normed space $\mathbb{R}$ (or $\mathbb{C}$) is a (continuous) linear functional on X. Recall that the space of all continuous linear functionals is denoted X ^∗ or $\mathop{\mathrm{B}}\nolimits (X, \mathbb{R})$ and it is called the dual or conjugate space of X. Lemma  2.54 shows that X ^∗ is a Banach space with respect to the operator norm.