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].