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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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That notation just means the set of all functions $f: X \longrightarrow R$. Look at the first example in this Wikipedia entry for a function space. –  Adrián Barquero Apr 11 '11 at 6:51
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Rajesh, I think you should edit the question's title because it has nothing to do with continuous functions or vector spaces. You're just asking about the notation. –  Adrián Barquero Apr 11 '11 at 6:58
    
read the first chunk of cantor's "contributions to the founding of the theory of transfinite numbers" –  yoyo Apr 11 '11 at 20:59
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up vote 3 down vote accepted

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 $Z^{(Y \times X)} = (Z^{Y})^{X}$. –  t.b. Apr 11 '11 at 7:08
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This means the space of all functions from $X$ to $R$. Without regard for any structure. Set-theoretic ones.

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