Predual

A predual space is a particular kind of dual space. To fully understand what a predual space is, we must first grasp the concept of a dual space and then look at preduals within that context.

Dual space

In the field of functional analysis, if $X$ is a normed vector space, then the dual space $X*$ is defined as the set of all continuous linear functionals from $X$ to its underlying field, which is usually the field of real or complex numbers. Formally, $X*$ consists of all mappings $\phi: X \rightarrow \mathbb{C}$ (if the field is complex numbers) such that for any $x$, $y$ in $X$ and any scalar $\alpha$, we have the following:

1. $\phi(x + y) = \phi(x) + \phi(y)$ (linearity)
2. $\phi(\alpha x) = \alpha \phi(x)$ (scalar homogeneity)
3. $|\phi(x)| \leq M|x|$ for some constant $M$ (continuity in the operator norm)

The set $X*$ equipped with the operator norm is a Banach space (a complete normed vector space).

Predual Space

The predual of a given normed space $Y$ is a normed space $X$ such that the dual of $X$ ($X*$) is isometrically isomorphic to $Y$. This is to say, there is a bijective mapping $T: X* \rightarrow Y$ such that $|Tx| = |x|$ for all $x$ in $X*$. This is a way of "embedding" $Y$ within the dual of $X$.

Predual spaces are not unique: a single normed space may have many different preduals. However, certain important spaces, like the space of bounded linear operators on a Hilbert space, do have unique preduals.

In the context of operator theory, preduals are often used to study properties of operators on Banach spaces. For example, if $Y$ is the space of bounded linear operators on some Hilbert space $H$, then the predual of $Y$ can be identified with the space of trace class operators on $H$, and this identification can be used to define a trace on $Y$.

Preduals are particularly important in the theory of von Neumann algebras. Every von Neumann algebra $M$ on a Hilbert space $H$ has a unique predual, which can be identified with the space of all normal states on $M$. This allows for a very rich structure theory of von Neumann algebras based on preduals.

It should be noted that this is a very advanced topic in mathematics and typically requires a good understanding of functional analysis and operator theory.