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Inequivalent representations of the dual space
Tepper L. Gill, Douglas Mupasiri, and Erdal Gül
Volume 64, no. 2
(2022),
pp. 271–280
https://doi.org/10.33044/revuma.3065
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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.
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