A lifting theorem for locally convex subspaces of $L_0$

ArticleOriginal scientific text

Title

Authors ¹

Department of Mathematics, University of Illinois, Urbana, Illinois 61801, U.S.A.

We prove that for every closed locally convex subspace E of

L_{0}

and for any continuous linear operator T from

L_{0}

to

\frac{L_{0}}{E}

there is a continuous linear operator S from

L_{0}

to

L_{0}

such that T = QS where Q is the quotient map from

L_{0}

to

\frac{L_{0}}{E}

.

T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), No. 10
N. J. Kalton and N. T. Peck, Quotients of $L_{p}$ for 0≤ p<1, Studia Math. 64 (1979), 65-75.
N. J. Kalton, N. T. Peck and J. W. Roberts, An F-space Sampler, Cambridge Univ. Press, Cambridge, 1984.
S. Kwapień, On the form of a linear operator in the space of all measurable functions, Bull. Acad. Polon. Sci. 21 (1973), 951-954.
R. E. A. C. Paley and A. Zygmund, On some series of functions III, Proc. Cambridge Philos. Soc. 28 (1932), 190-205.
N. T. Peck and T. Starbird, $L_{0}$ is ω-transitive, Proc. Amer. Math. Soc. 83 (1981), 700-704.
S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. 5 (1957), 471-473.
P. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math. 131 (1976).