How are these notations related (if at all), or are they the same? Since I'm not sure if it's what people usually write, I'll clarify $\langle,\rangle$ is a mapping.
- $\langle a,b\rangle=c$
- $a:b\to c$
- $a(b)=c$ (a as a function of b)
- $\langle,\rangle:(a,b)\to c$ or $\langle,\rangle:a\times b\to c$
I'm also not sure what any of these mean, so there could be mistakes. Some of the notation I've used here is from a youtube channel XylyXylyX, what is a tensor.
Edits: I think 2 and 3 are the same, but someone told me 2 and 1 mean the same thing and $a(b)$ is just one type of way to make $a$ be a map (act on something). Or (as in 4) is $\langle,\rangle$ the map but information about it is encoded in $a$ ($a$ is just used somewhere but is part of the function like $\langle a,b\rangle=ab$)?
Technical changes: I added "if at all" to the first sentence and explicitly stated I'm using $\langle,\rangle$ because that's what I'm used to, even though it's also sometimes used as an inner product. Since this may have made the question misleading I changed the title from "Notations for Mappings" to "What Notations Describe a Mapping". Hopefully these flush out the question a little more.