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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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    read the first chunk of cantor's "contributions to the founding of the theory of transfinite numbers"2011-04-11

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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.

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    @Rajesh: Further motivation can be seen in the so-called *exponential law* $Z^{(Y \times X)} = (Z^{Y})^{X}$.2011-04-11
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This means the space of all functions from $X$ to $R$. Without regard for any structure. Set-theoretic ones.