Let $V$ be a vector space over $\mathbb R$. The algebraic dual of $V$, denoted $V^*$, is the space of all linear functions from $V$ to $\mathbb R$. These functions are called functionals. Most textbooks list the following as examples for functionals:
- The integration operator $f\mapsto \int f$ is a functional.
- The evaluation functional (at the query point $x$) $f\mapsto f(x)$
Obviously, both examples are only meaningful in the case $V$ is a function space of real-valued functions, e.g. $L^p(X,\mathbb R)$, or $C^p(X,\mathbb R)$.
Now, suppose we are looking at spaces of functions which map the domain $X$ to, say, $\mathbb R^k$, or even more general to an arbitrary (Banach) spaces $B$. The integral operator remains a functional as integration, by definition, yields eventually a real number. However, the evaluation functional is no longer a functional as the function in place no longer takes values in $\mathbb R$, right? So I was wondering whether an evaluation "functional" is also defined for such cases? Or do I get something fundamentally wrong?