Banach–Mazur theorem

On the one hand, the Banach–Mazur theorem seems to tell us that the seemingly vast collection of all separable Banach spaces is not that vast or difficult to work with, since a separable Banach space is "only" a collection of continuous paths.The embedding j is introduced by saying that for every x ∈ X, the continuous function j(x) on K is defined by The mapping j is linear, and it is isometric by the Hahn–Banach theorem.In 1995, Luis Rodríguez-Piazza proved that the isometry i : X → C0[0, 1] can be chosen so that every non-zero function in the image i(X) is nowhere differentiable.This conclusion applies to the space C0[0, 1] itself, hence there exists a linear map i : C0[0, 1] → C0[0, 1] that is an isometry onto its image, such that image under i of C0[0, 1] (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects D only at 0: thus the space of smooth functions (with respect to the uniform distance) is isometrically isomorphic to a space of nowhere-differentiable functions.Note that the (metrically incomplete) space of smooth functions is dense in C0[0, 1].
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