C1-fine approximation of functions on Banach spaces with unconditional basis

Abstract

We show that if X is a Banach space having an unconditional basis and a C p -smooth Lipschitz bump function, then for every C 1 -smooth function f from X into a Banach space Y , and for every continuous function ε : X → (0, ∞), there exists a C p -smooth function g : X → Y such that f − g ε and f − g ε.

Smooth extensions of functions on separable Banach spaces

“…We modify the proof of Theorem 4 in [4] employing Lemma 1. It will be convenient to use the following notation: given a point y k ∈ Y, we define T k to be the natural H -extension of the first order Taylor Polynomial of f at y k ; namely,…”

“…We have combined some recent work on smooth approximation of Lipschitz mappings [14] with some techniques from Moulis [18] and [4], the Bartle-Graves selector theorem, and the classical method of Tietze to deduce our principal results.…”

Approximation by Cp-smooth, Lipschitz functions on Banach spaces

“…The proof of our main theorem is a modification of some techniques found in [2,11], where C p -fine approximation on Banach spaces is considered. We include here the main parts of the construction, and shall refer the reader to [2] in the appropriate places to avoid excessive technical detail. Theorem 1.…”

“…Also, since (χ n (x)x n ) = χ n (x)x n + χ n (x)e * n , it follows that Ψ (x)(·) = n χ n (x)(·)x n e n + n χ n (x)e n e * n (·). (2.4) Now, using (2.4), the estimate for χ n above, and again using the fact that {e n } is unconditional with constant C, it is shown in [2,Fact 7] that for all x ∈ X, Ψ (x)…”

Approximation by smooth functions with no critical points on separable Banach spaces

“…We will use the notation H k and H k when the function Since the norm | · | is LUR we can find, for every x ∈ S + , open slices R x = {y ∈ S: f x (y) > δ x } ⊂ S + and P x = {y ∈ S: f x (y) > δ 4 x } ⊂ S + , where 0 < δ x < 1 and |f x | = 1 = f x (x), so that the oscillation of F in every P x is less than ε. We also assume, for technical reasons, and with no loss of generality, that dist(P x , X × {0}) > 0.…”