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Non‐linear factorization of linear operators
Author(s) -
Johnson W. B.,
Maurey B.,
Schechtman G.
Publication year - 2009
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp040
Subject(s) - mathematics , lipschitz continuity , factorization , banach space , linearization , simple (philosophy) , linear operators , pure mathematics , operator (biology) , linear map , discrete mathematics , mathematical analysis , nonlinear system , algorithm , philosophy , biochemistry , physics , chemistry , epistemology , quantum mechanics , repressor , transcription factor , gene , bounded function
We show, in particular, that a linear operator between finite‐dimensional normed spaces, which factors through a third Banach space Z via Lipschitz maps, factors linearly through the identity from L ∞ ([0, 1], Z ) to L 1 ([0, 1], Z ) (and thus, in particular, through each L p ( Z ), for 1 ⩽ p ⩽ ∞) with the same factorization constant. It follows that, for each 1 ⩽ p ⩽ ∞, the class of ℒ p spaces is closed under uniform (and even coarse) equivalences. The case p = 1 is new and solves a problem raised by Heinrich and Mankiewicz in 1982. The proof is based on a simple local–global linearization idea.