Rudin PMA p.161
It says, A family $\mathscr{A}$ of complex functions defined on a set $E$ is said to be an algebra if (i) $f+g\in\mathscr{A}$, (ii)$fg\in\mathscr{A}$, and (iii) $cf\in\mathscr{A}$ for all $f\in\mathscr{A}, g\in\mathscr{A}, c\in\mathbb{C}$.
I guess it is definitely not a definition of 'algebra' generally, but it is an example of 'algebra'. And how do i call it? $\mathscr{A}$ is an algebra on $E$?
I'm asking this because it seems Rudin doesn't make definitions very clearly.
(He even defined 'equicontinuous' as the definition of 'uniformly equicontinuous' on wikipedia, which are totally different)
I tried to understand definition on wikipedia myself, but then there are many terms i'm not familiar with, so i tried to find meanings of those terms, then again there are terms i don't know.
So far, i saw that algebra has different meanings depend on situation, and don't even know what 'algebra' here in my book means.
Please someone explain what kind of algebra it is, and what is the definition of that algebra in relatively easy terms.
Thank you in advance