Inequivalent representations of the dual space

Abstract

We show that there exist inequivalent representations of the dual space of $\mathbb{C}[0,1]$ and of $L_p[\mathbb{R}^n]$ for $p \in [1,\infty)$. We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if $\mathcal{A}$ is a proper closed subspace of $\mathbb{C}[0,1]$, there always exists a closed subspace $\mathcal{A}^\bot$ (with the same definition as for $L_2[0,1]$) such that $\mathcal{A} \cap\mathcal{A}^\bot = \{0\}$ and $\mathcal{A} \oplus \mathcal{A}^\bot =\mathbb{C}[0,1]$. Thus, the geometry of $\mathbb{C}[0,1]$ can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator $A$ on $\mathbb{C}[0,1]$ has a uniquely defined adjoint $A^*$ defined on $\mathbb{C}[0,1]$, and both can be extended to bounded linear operators on $L_2[0,1]$. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to $L_p[\mathbb{R}^n]$. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on $\mathbb{C}[0,1]$ is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.