how to understand the (f+g)(x) and (αf)(x) notation In the following notations:
$$ (f+g)(x)=f(x)+g(x) $$
$$ (αf+βg)(x)=αf(x)+βg(x) $$
can anyone help to understand:
$$(f+g)(x) $$
$$ (αf+βg)(x) $$
?
It's no problem if $f$ and $g$ are linear transformation matrices $A$ and $B$ and $x$ is a vector variable $X$ because I can compute and compare $(A+B)X$ and $AX+BX$.
Otherwise, when it's non-linear, the notation is hard to understand.
for example,
$$ f(x)=\cos x+1 $$
$$ g(x)=e^x-2 $$
I can write:
$$ f(x)+g(x)=\cos x+1 + e^x-2 $$
$$ αf(x)+βg(x) = α\cdot \cos x+α + β\cdot e^x -2β$$
In this case, what are the definition of (and how to evaluate):
$$(f+g)(x) $$
$$ (αf+βg)(x) $$
?
I'm asking this in order to understand the definition of dual space (en.wikipedia.org/wiki/Dual_space) which suggests $(f+g)(x)=f(x)+g(x)$ and $(αf+βg)(x)=αf(x)+βg(x)$ are to be proved/verified (note the word "satisfying" the conditions)
 A: $f+g$ is the function defined by
$$
(f+g)(x) = f(x) + g(x)
$$
for all $x$. So '$f+g$' is just a name we give to a function—but it is a meaningful name, since when people talk of the function '$f+g$' I immediately know what function they are talking about. But if you don't like this notation, then you could just call this function something else instead. E.g. you could let $h$ be the function defined by
$$
h(x)=f(x)+g(x) \, .
$$
It's important to distinguish between a function and its values: $f$ is the function, whereas $f(x)$ is the function evaluated at an arbitrary point $x$. When people speak of 'the function $f(x)$', you should keep in mind that this is just an imprecise yet convenient shorthand. It's the same story with the function $f+g$. The function $f+g$ has the property that when it is evaluated at an arbitrary point $x$,
$$
(f+g)(x)=f(x)+g(x) \, .
$$
Hopefully from this you can guess the meaning of $f \cdot g$ as well: it's the function with the property that
$$
(f \cdot g)(x) = f(x) \cdot g(x) \, .
$$
Sometimes we write simply $fg$ to mean $f \cdot g$, but this notation can be ambiguous since $fg$ can also denote the composite function $f \circ g$.
A: This is a typical way to define an operation on a space of functions "point-wise". A function $f:X\to Y$ between two sets $X$ and $Y$ is given by $\{ (x,f(x)) : x\in X \}$; i.e. by giving its values at every point $x$. Now given two functions $f:X\to Y$ and $g:X\to Y$, you may define
$$
(\alpha\cdot f + \beta\cdot g) :=\{ (x, \alpha\cdot f(x) + \beta\cdot g(x)):x\in X\}
$$
which is precisely what is meant by saying that $\alpha\cdot f + \beta\cdot g$ is defined as
$$
(\alpha\cdot f + \beta\cdot g)(x) := \alpha\cdot f(x) + \beta\cdot g(x).
$$
The above notation is slightly abusive because the quantifier is left out; it is implicit. The definition intends to say that you make this definition for every $x\in X$, thereby defining the whole function.
More generally, if you have any $k$-ary operation $\omega: Y^k\to Y$ on a set $Y$, you can extend this operation to the space of maps $Y^X$ for any set $X$. Indeed, for $f_1,\ldots,f_k\in Y^X$ you define
$$
\omega(f_1,\ldots,f_k) := \{ (x, \omega(f_1(x),\ldots,f_k(x)) : x\in X \}
$$
This is referred to as defining $\omega$ point-wise on $Y^X$.
