Positivity 2: 339–367, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Subspaces of L
Isometric to Subspaces of
Department Mathematik, ETH Zürich, Rämistr 101, CH 8092 Zürich, Switzerland
Institut für Mathematik, Universität Zürich, Winterthurerstr 190, CH 8057 Zürich, Switzerland
Institute of Mathematics, Polish Academy of Sciences, 8, ul.Sniadeckich, PL 00-950 Warszawa,
(Received 11 May 1998; Accepted: 11 May 1998)
Abstract. We present three results on isometric embeddings of a (closed, linear) subspace X of L
[0, 1] into
. First we show that if p/∈ 2
,thenX is isometrically isomorphic to a subspace of
if and only if some, equivalently every, subspace of L
which contains the constant functions
and which is isometrically isomorphic to X, consists of functions having discrete distribution. In
contrast, if p ∈ 2
and X is ﬁnite-dimensional, then X is isometrically isomorphic to a subspace of
, where the positive integer N depends on the dimension of X,onp, and on the chosen scalar
ﬁeld. The third result, stated in local terms, shows in particular that if p is not an even integer, then
no ﬁnite-dimensional Banach space can be isometrically universal for the 2-dimensional subspaces
Mathematics Subject Classiﬁcations (1991): Primary: 46B04, 46E30; Secondary: 46B25, 46B07.
Key words: Subspaces of L
, linear isometries, distribution measures, equimeasurable func-
tions, Banach-Mazur compactum, isometrically universal ﬁnite-dimensional Banach spaces
STATEMENT OF RESULTS
The starting point of the present paper was a question of Albrecht Pietsch: “Which
ﬁnite-dimensional subspaces of L
are isometrically isomorphic to subspaces of
?" The purpose of this note is to give a comprehensive answer to Pietsch’s
question. Curiously, the answer depends on the arithmetic nature of p and splits
into two cases. In the ﬁrst case p is an arbitrary positive number but not an even
integer, in the second case p is a positive even integer (in symbols p∈2
Supported by the Swiss National Funds.
Supported by KBN grant 2P0 3A 03614.