Complementary Subspaces

2014 ◽  
pp. 173-196
Keyword(s):  
Complementary Subspaces

A Note on Complementary Subspaces in c0

1972 ◽  
Vol 24 (3) ◽  
pp. 537-540
Author(s):  
I. D. Berg
Keyword(s):  
Usual Basis ◽  
Complementary Subspace ◽  
Complementary Subspaces ◽  
Basis Vectors

A well known result of A. Pelcynski [2] states that each subspace of c0 which is isomorphic to c0 and of infinite deficiency has a complementary subspace which is itself isomorphic to c0. We are concerned here with the question of when there exists R, a subset of the integers, such that the complementary subspace X can actually be taken to be C0(R). That is, we are concerned with determining when the basis vectors for X can be chosen as a subset of the usual basis vectors for c0. If T: C0 → C0 is norm increasing and ‖T‖ < 2, it is not hard to see, as we shall show, that Tco admits a complement of the form C0(R). However, this bound cannot be improved; indeed, it is possible to construct norm increasing T: C0 → C0 such that ‖T‖ = 2 and yet Tc0 admits no such complement. The construction of such a T is the main point of this note. This construction also enables us to dispose of a speculation of ours in [1].

scholarly journals On complementary subspaces of Hilbert space

1998 ◽  
Vol 126 (10) ◽  
pp. 3019-3026 ◽  
Author(s):  
W. E. Longstaff ◽  
Oreste Panaia
Keyword(s):  
Hilbert Space ◽  
Complementary Subspaces

The Angle Between Complementary Subspaces

1995 ◽  
Vol 102 (10) ◽  
pp. 904 ◽  
Author(s):  
Ilse C. F. Ipsen ◽  
Carl D. Meyer
Keyword(s):  
Complementary Subspaces

scholarly journals The fit and flat components of barrelled spaces

1995 ◽  
Vol 51 (3) ◽  
pp. 521-528 ◽  
Author(s):  
Stephen A. Saxon ◽  
Ian Tweddle
Keyword(s):  
Continuum Hypothesis ◽  
Dense Subspace ◽  
Splitting Theorem ◽  
Hamel Basis ◽  
Barrelled Spaces ◽  
Barrelled Space ◽  
Complementary Subspaces ◽  
The Continuum

The Splitting Theorem says that any given Hamel basis for a (Hausdorff) barrelled space E determines topologically complementary subspaces Ec and ED, and that Ec is flat, that is, contains no proper dense subspace. By use of this device it was shown that if E is non-flat it must contain a dense subspace of codimension at least ℵ0; here we maximally increase the estimate to ℵ1 under the assumption that the dominating cardinal ∂ equals ℵ1 [strictly weaker than the Continuum Hypothesis (CH)]. A related assumption strictly weaker than the Generalised CH allows us to prove that ED is fit, that is, contains a dense subspace whose codimension in ED is dim (ED), the largest imaginable. Thus the two components are extreme opposites, and E itself is fit if and only if dim (ED) ≥ dim (Ec), in which case there is a choice of basis for which ED = E. Morover, E is non-flat (if and) only if the codimension of E′ is at least in E*. These results ensure latitude in the search for certain subspaces of E* transverse to E′, as in the barrelled countable enlargement (BCE) problem, and show that every non-flat GM-space has a BCE.

scholarly journals Construction of caps by means of caps in complementary subspaces

2009 ◽  
Vol 309 (2) ◽  
pp. 497-500 ◽  
Author(s):  
Hans-Joachim Kroll ◽  
Rita Vincenti
Keyword(s):  
Complementary Subspaces

Classification by nearness in complementary subspaces

2012 ◽  
Vol 16 (4) ◽  
pp. 609-622
Author(s):  
Menglong Yang ◽  
Yiguang Liu ◽  
Baojiang Zhong ◽  
Zheng Li
Keyword(s):  
Complementary Subspaces

scholarly journals On the Finite Dimensionality of Closed Subspaces in Lp(M,dμ)∩Lq(M,dν)

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 275
Author(s):  
Alexander A. Balinsky ◽  
Anatolij K. Prykarpatski
Keyword(s):  
Banach Space ◽  
Probability Measure ◽  
Closed Subspace ◽  
A Priori ◽  
Functional Structure ◽  
Functional Constraints ◽  
Interesting Problem ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Complementary Subspaces

Finding effective finite-dimensional criteria for closed subspaces in Lp, endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufficient constraints on closed functional subspaces, Sp⊂Lp, whose finite dimensionality is not fixed a priori and can not be checked directly. This is often the case in diverse applications, when a closed subspace Sp⊂Lp is constructed by means of some additional conditions and constraints on Lp with no direct exemplification of the functional structure of its elements. We consider a closed topological subspace, Sp(q), of the functional Banach space, Lp(M,dμ), and, moreover, one assumes that additionally, Sp(q)⊂Lq(M,dν) is subject to a probability measure ν on M. Then, we show that closed subspaces of Lp(M,dμ)∩Lq(M,dν) for q>max{1,p},p>0 are finite dimensional. The finite dimensionality result concerning the case when q>p>0 is open and needs more sophisticated techniques, mainly based on analysis of the complementary subspaces to Lp(M,dμ)∩Lq(M,dν).

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