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I have never used this notation before. I did some digging, but closest thing I could find is: What does the function f: x ↦ y mean?

Can anybody explain this notation?

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It means that $H$ is the set of all functions from the set $X$ to the set $Y$.

Read literally, we could say the statement as "$H$ is the set of all functions $h$ such that $h$ has domain $X$ and codomain $Y$."

(Anecdotally, $X,Y$ need not necessarily be sets - could be rings, vector spaces, groups, whatever: the context in which this appears should make it clear. I imagine if this is your first time coming across it, it's probably about sets though.)

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$H$ is the set of all functions with domain $X$ and codomain $Y$.

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Another way to write it would be $H=Y^X$. It's the set of all functions from $X$ to $Y$.

I believe this notation is used in functional analysis.

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