a question on notation for function spaces
If $X$ is some topological space, such as the unit interval $[0,1]$, we can consider the space of all continuous functions from $X$ to $R$. This is a vector subspace of $R^X$ since the sum of any two continuous functions is continuous and scalar multiplication is continuous.
Please let me know the notation $R^X$ in the above example.
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$\begingroup$To see the motivation for this notation (and thus also to remember it more easily), note that $|A^X|=|A|^{|X|}$. The analogy with exponentiation is even more direct if we use the set-theoretic construction of a natural number as the set of all its predecessors, e.g. $3=\{0,1,2\}$. In that case the sets denoted by $A^n$ (the $n$-fold Cartesian product of $A$, and $A^X$ with $X=n$ in the above sense) are isomorphic; in fact, under a set-theoretic definition of the Cartesian product, they are the same thing.
$\endgroup$ 1 $\begingroup$This means the space of all functions from $X$ to $R$. Without regard for any structure. Set-theoretic ones.
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