I'm reading the textbook "Monotone Operators in Banach Space and Nonlinear Partial Differential Equation" and I found a term that I'm not familiar with and I can't find a proper definition.
Let $X$ be a reflexive Banach space with dual $X'$, $H$ a Hilbert space such that $X$ is dense and continuously embedded in $H$. Suppose we are given a (not necessarily linear) function $A: X\to X'$. $A$ is known to map $L^2(0,T;X)$ into $L^2(0,T;X')$, i.e., the realization of $A: X\to X'$ as an operator on $L^2(0,T;X)$ has values in $L^2(0,T;X')$.
What is a "realization"?
The map $A$ has domain $X$ and codomain $X'$, so it is not a map from $L^2([0,T],X)$ to $L^2([0,T],X')$. But it does induce such a map naturally, by setting $\tilde {A}(\phi)=A\circ \phi$ where $\phi\in L^2([0,T],X)$.
The author simply does not want to bother using a different notation for $A$ and $\tilde A$. Note that the actual point of the author is that the map $A\circ \phi$ lies in $L^2$ as well.