What type of Banach spaces $X$ does the sum $x + c$ make sense where $x \in X$ and $c \in \mathbb{R}$?

Question

What are such spaces called where we can add a constant to an element of the Banach space and the addition makes sense somehow?

Eg. in $L^2$ this always is sensible. Is there a difference to the name if $x+c \notin X$ to $x+c \in X$?

Answer 1

At face value, this never has any meaning unless $X$ is a field containing real numbers as a subfield. But, of course, as you mentioned, one can assign meaning to things like adding $c$ to a function, by defining $c$ not to be an element of $\mathbb{R}$, but an element of the set of constant functions.

This is always possible, for example, if $X$ is a Banach space which also has a unital algebra structure (e.g. a unital Banach algebra). Then $X$ has a multiplicative unit, which we denote $1_X$, and $c := c 1_X$, in which case one can define $x + c$. This is the case in spaces such as, for example, $C(\Omega)$, where $\Omega$ is a compact set.