In mathematics, the Banach–Mazur theorem is a theorem of functional analysis. Very roughly, it states that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named after Stefan Banach and Stanisław Mazur.
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Every real, separable Banach space (X, || ||) is isometrically isomorphic to a closed subspace of C 0([0, 1]; R), the space of all continuous functions from the unit interval into the real line.
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 "just" a collection of continuous paths. On the other hand, the theorem tells us that C 0([0, 1]; R) is a "really big" space, big enough to contain every possible separable Banach space.
Let's write C k[0, 1] for C k([0, 1]; R). In 1995, Luis Rodríguez-Piazza proved that the isometry i : X → C 0[0, 1] can be chosen so that every non-zero function in the image i(X) is nowhere differentiable. Put another way, if D denotes the subset of C 0[0, 1] consisting of those functions that are differentiable at least one point of [0, 1], then i can be chosen so that i(X) ∩ D = {0}. This conclusion applies to the space C 0[0, 1] itself, hence there exists a linear map i from C 0[0, 1] to itself that is an isometry onto its image, such that image under i of C 1[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 (w.r. 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 C 0[0, 1].