Can a linear functional be infinite at a point? On a Banach (or Hilbert) space $X$, when we define a linear functional (not necessarily bounded), we define it to be a linear function from the elements of $X$ to the field $\Bbb F$. (Say, $\Bbb R$). Does that definition admit $\infty$ as a value for a certain $x \in X$? Going by the definition, it does not seem so, as I don't think that $\infty \in \Bbb R$ formally. But the Minkowsky functional, for instance, does admit $\infty$ as a value. Is that a misnomer, and it's not really a functional? For a concrete example, can I say that tht $f(x) = \sum_1^\infty x_i$ is a linear functional on $l^2$?

The value infinity is not allowed. Your example functional $$ f(x) = \sum_{i=1}^\infty x_i $$ is not defined on the whole $l^2$, $D(f)\subsetneq l^2$.

Unbounded functionals are somewhat different: they are defined on the whole space, but are unbounded. In every neighborhood of the origin there are points, where the unbounded functional admits arbitrarily large values.