By Joram Lindenstrauss
This publication makes an important inroad into the without warning tough query of lifestyles of Frchet derivatives of Lipschitz maps of Banach areas into greater dimensional areas. as the query seems to be heavily relating to porous units in Banach areas, it offers a bridge among descriptive set conception and the classical subject of lifestyles of derivatives of vector-valued Lipschitz capabilities. the subject is appropriate to classical research and descriptive set idea on Banach areas. The publication opens a number of new learn instructions during this quarter of geometric nonlinear practical research. the recent equipment constructed right here comprise a online game method of perturbational variational ideas that's of autonomous curiosity. targeted rationalization of the underlying principles and motivation at the back of the proofs of the hot effects on Frchet differentiability of vector-valued services should still make those arguments obtainable to a much broader viewers. an important specified case of the differentiability effects, that Lipschitz mappings from a Hilbert area into the aircraft have issues of Frchet differentiability, is given its personal bankruptcy with an evidence that's self sustaining of a lot of the paintings performed to end up extra normal effects. The publication increases numerous open questions bearing on its major issues.
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Extra info for Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces
Proof. Let x ∈ X with x = 1. Find x∗ ∈ X ∗ such that x∗ = x∗ (x) = 1. For every u ∈ X we choose a norm one functional yu∗ in X ∗ such that yu∗ (x+u) = x+u . Then yu∗ + x∗ ≥ yu∗ (x) + x∗ (x) = x + u − yu∗ (u) + 1. Hence yu∗ + x∗ → 2 as u → 0. By the local uniform convexity of X ∗ we deduce that yu∗ − x∗ → 0 as u → 0. It follows that ∗ 0 ≤ x + u + x − u − 2 = yu∗ (x + u) + y−u (x − u) − 2 ∗ ∗ = yu∗ (x) + y−u (x) − 2 + (yu∗ − y−u )(u) ∗ ≤ yu∗ − y−u u = o( u ). 1. Since both X and X ∗ are separable there are a dense sequence (xn )∞ n=1 in the unit sphere of X and an increasing sequence of finite dimensional sub∗ spaces (Fn )∞ n=1 of X such that X∗ = Fn .
This is enough for our purposes, but let us note that we will actually use only the following consequence of uniform equicontinuity. For every ε > 0 there is δ > 0 such that whenever F ∈ F(U ), u, v ∈ U p , y ∈ Y p , and u − v < δ, then |F (u, y) − F (v, y)| < ε. 3. If ε and δ are as above and W is a subspace of X, then for every x, x ˜ ∈ W with x − x ˜ < δ, β(f, x, W, F) ≤ β(f, x ˜, W, F) + ε. Proof. Fix any 0 < c < β(f, x, W, F). By definition of β(f, x, W, F) there is a separable subspace U ⊂ W such that β(f, x, U, F ) > c for every F ∈ F(U ).
F ) + C 2 + Cy ∗ (f (x)(e) − w) > C 2 + C − 2, implying that y ∗ (f (x)(e) − w)) > − 2 (1 + Lip C · (f )), which, multiplied by b − a and for C large enough, gives the statement. So far we have considered what could be termed one-dimensional mean value estimates: although the range could be even infinite dimensional, the estimate involved only derivative in a single direction. 3). The natural formulation may then sound somewhat awkward, but we may follow the same reasoning as above. If, instead of a line segment, we imagine an n-dimensional (say C 1 ) surface in X, then for any Lipschitz f : X −→ Y and u∗1 , .