Of course there are uncomplimented subspaces of Banach spaces but the sequence space little-Lp is "projective" for the category of p-Banach spaces. Thus, if X is a p-Banach space with closed subspace N and T is an operator from little-Lp to X/N there is a lifting S(not necessarily unique) from little-Lp to X.
Moreover, since X/N is p-normable there is an index set I such that there is a surjection from little-Lp(I) to X/N. Thus, this surjection has a lifting to X. Does this imply N is complimented or is the uniqueness of the lifting required?
Dear Geoff,
In Lp[0,1] with p < 1 there is no continuous non-zero linear functional, consequently there is no complemented 1-dimensional subspace.
Best regards, Vladimir
It is interesting to note that L_p(X) of a finite measure space has a non zero continuous linear functional iff X has an atom. In fact, atoms in X and such functionals are in one one correspondance.
Dear Geoff, Vladimir and Subhash, I know (nonconstructive) examples of noncomplemented closed subspaces of l_p ("l" small, p
Dear Oleg,
I don't know such examples, but this does not say much: I never worked with \ell_p with p < 1, and in general know very little about not locally convex spaces. Maybe people that work with p-Banach spaces know something about this. Unfortunately, we cannot ask advise from Nigel Kalton any more ...
Best regards, Vladimir
Try searching around through these web sites to see if it helps. Wish I could do more but not to familiar with it, either.
http://math.stackexchange.com/questions/1928771/ell-p-spaces-differences-between-p-and-p
or
https://www.researchgate.net/publication/2299057_Well-Posedness_of_a_Dissolution-Growth_Problem_on_an_Unbounded_Domain
May try also searching the question within research gate as well there may be some one on here that has worked with this application before.
Good Luck in research.
Yes, Nigel Kalton has been a great loss to mathematics. Not only because he seemed to know everything but he usually had time to talk about it.
See e.g.
KALTON, N. J. “The Endomorphisms of L_p (0 ≤ p ≤ 1). Indiana University Mathematics Journal, vol. 27, no. 3, 1978, pp. 353–381.,
www.jstor.org/stable/24892314,
Corillary 4.5 and a note after its proof:
"In particular there is no projection onto L_p [[0, 1] x [0, 1]] onto its subspace
of all functions constant on horizontal lines if 0 ≤ p < 1"
Every infinite-dimensional complemented subspace of l_p is isomorphic to l_p (Stiles). In other words, any infinite-dimensional subspace of l_p not isomorphic to l_p is uncomplemented. Several examples appeared in the literature (see e.g. [Kalton, Banach envelopes of quasi-Banach spaces, Canadien JM], [Kalton, script-Lp spaces for p
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