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By appropriate choices of elements in the underlying iterated functionsystem, methodology of fractal interpolation entitles one to associate a familyof continuous self-referential functions with a prescribed real-valuedcontinuous function on a real compact interval. This procedure elicits what isreferred to as $\alpha$-fractal operator on $\mathcal{C}(I)$, the space of allreal-valued continuous functions defined on a compact interval $I$. With an eyetowards connecting fractal functions with other branches of mathematics, inthis article, we continue to investigate fractal operator in more generalspaces such as the space $\mathcal{B}(I)$ of all bounded functions and Lebesguespace $\mathcal{L}^p(I)$, and some standard spaces of smooth functions such asthe space $\mathcal{C}^k(I)$ of $k$-times continuously differentiablefunctions, H\"{o}lder spaces $\mathcal{C}^{k,\sigma}(I)$, and Sobolev spaces$\mathcal{W}^{k,p}(I)$. Using properties of the $\alpha$-fractal operator, theexistence of Schauder basis consisting of self-referential functions isestablished.

A Fractal Operator on Some Standard Spaces of Functions