# Predual

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{{short description|Banach space of a dual}}
{{One source|date=March 2026}}
In [mathematics](/source/mathematics), the '''predual''' of an object ''D'' is an object ''P'' whose [dual space](/source/dual_space) is ''D''.

For example, the predual of the space of [bounded operator](/source/bounded_operator)s is the space of [trace class](/source/trace_class) operators, and the predual of the space ''L''<sup>∞</sup>('''R''') of essentially bounded functions on '''R''' is the Banach space ''L''<sup>1</sup>('''R''') of integrable functions.

In [operator algebra](/source/operator_algebra), if a dual Banach/operator space <math>A</math> is realized as the dual of some [Banach space](/source/Banach_space) <math>A_*</math>, then 
<math>A_*</math> is called the predual of <math>A</math> (Formally: <math>A \cong (A_* )^*</math>)
The predual <math>A_*</math> induces a [weak topology](/source/weak_topology) on <math>A</math>, under which [algebra operations](/source/operator_algebra) are separately weak continuous.<ref>{{cite journal
 | last = Ruan
 | first = Zhong-Jin
 | year = 1992
 | title = On the predual of dual algebras
 | journal = Journal of Operator Theory
 | volume = 27
 | issue = 1
 | pages = 179–192
 | doi = 10.2307/24715083
}}</ref>

<!-- other, simpler, interesting examples? -->

==References==
{{reflist}}

Category:Abstract algebra
Category:Functional analysis

{{mathanalysis-stub}}

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