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The well-known Banach-Mazur theorem says that every separable Banach space can be isometrically embedded into C([0, 1]). We prove that this embedding can have the property that the image of each nonzero element is a nowhere approximatively differentiable and nowhere Holder function. It improves a recent result of L. Rodriguez-Piazza where the images are nowhere differentiable functions. Introduction The well-known Banach-Mazur theorem says that every separable Banach space can be isometrically embedded into C([0, 1]). We prove that this embedding can have the property that the image of each nonzero element is a nowhere approximatively differentiable and nowhere Holder function. It improves a recent result of L.Rodriguez-Piazza [2] where the images are nowhere differentiable functions (for references about results concerning embeddings into subsets of C([0, 1]) see [2]). The basic idea of our proof is the same as in [2] but we use a more complicated construction which uses an idea of Malý and Zajicek [1]. Let ∆ be the Cantor set. It is well known that every separable Banach space is isometric to a subspace of C(∆) so the following theorem will be enough to obtain the announced result. Theorem 1. There exist a closed subset K of [0, 1] homeomorphic to the Cantor set ∆ and a linear operator F : C(K)→ C([0, 1]) such that for every f ∈ C(K)\{0} we have: (i) Ff(t) = f(t) for every t ∈ K, so Ff is a continuous extension of f to the whole interval. (ii) ‖ f ‖∞ = ‖ Ff ‖∞, so Ff is an isometry. (iii) Ff is nowhere approximatively differentiable and nowhere Holder function. In fact we will prove a stronger result: Proposition 2. Let φ : [0,∞) → [0,∞) be a continuous increasing function such that φ(0) = 0. Then there exist a closed subset K of [0, 1] homeomorphic to the Cantor set ∆ and a linear operator F : C(K) → C([0, 1]) such that for every f ∈ C(K) \ {0} we have: Received by the editors January 22, 1999. 1991 Mathematics Subject Classification. Primary 26A27, 46B04. c ©2000 American Mathematical Society

Isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions