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### Linearity and Complements in Projective Space [article]

Michael Braun, Tuvi Etzion, Alexander Vardy
2011 arXiv   pre-print
This motivates new interest in such codes. In this paper, we examine the two fundamental concepts of complements and linear codes in the context of .  ...  Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for error-correction in networks: an (n,M,d) code can correct t packet errors and ρ packet erasures introduced  ...  The complete answer is summarized in Table 2 . Conclusion and Open Problems We have considered various problems related to linear codes and complements in projective space.  ...

### Linearity and complements in projective space

Michael Braun, Tuvi Etzion, Alexander Vardy
2013 Linear Algebra and its Applications
This motivates our interest in such codes. In this paper, we examine the two fundamental concepts of "complements" and "linear codes" in the context of P q (n).  ...  Our results reveal a number of surprising phenomena pertaining to complements and linearity in P q (n) and gives rise to several interesting problems.  ...  The complete answer is summarized in Table 2 . Conclusion and open problems We have considered various problems related to linear codes and complements in projective space.  ...

### Projections in the space \$(m)\$

Robert C. James
1955 Proceedings of the American Mathematical Society
A projection in a Banach space is a continuous linear mapping P of the space into itself which is such that P2=P.  ...  Two closed linear manifolds M and N of a Banach space B are said to be complementary if each z of B is uniquely representable as x+y, where x is in M and y in N.  ...  PROJECTIONS IN THE SPACE (m)1 ROBERT C. JAMES A projection in a Banach space is a continuous linear mapping P of the space into itself which is such that P2=P.  ...

### \$1\$-complemented subspaces of spaces with \$1\$-unconditional bases

Beata Randrianantoanina
We completely characterize 1-complemented subspaces and norm-one projections in complex spaces ‡ p ( ‡ q ) for 1 pÒ q Ú 1.  ...  In particular if an Orlicz or Lorentz space X is not isomorphic to ‡ p for some 1 p Ú 1 then the only subspaces of X which are 1-complemented and disjointly supported are the closed linear spans of block  ...  We completely characterize 1-complemented subspaces and norm-one projections in complex spaces ‡ p ( ‡ q ) for 1 pÒ q Ú 1.  ...

### Projections in the spaces of bounded linear operations

Tsang Hai Kuo
1974 Pacific Journal of Mathematics
A subspace Y of a Banach space X is complemented if there is a projection P: X->X with range Y, i.e., a bounded linear operator of X such that P This conjecture was first studied by Thorp in , where  ...  For Banach spaces X, Z, let B(X, Z) denote the space of bounded linear operators from X into Z and K(X, Z) (resp. W(X, Z)) the subspace of compact (resp. weakly compact) operators.  ...  (a) Z is complemented in Z** if and only if Z is isomorphic to a complemented subspace of a dual space, (b) A bounded linear operator Te B(Y, Z) is weakly compact if and only if T** maps Γ** into Z, i.e  ...

### Projections in the Space (m)

Robert C. James
1955 Proceedings of the American Mathematical Society
JAMES A projection in a Banach space is a continuous linear mapping P of the space into itself which is such that P?=P.  ...  Curtiss, “Monte Carlo” methods for the iteration of linear operators, Journal of Mathematics and Physics vol. 32 (1954) pp. 209-232. PRINCETON UNIVERSITY PROJECTIONS IN THE SPACE (m)! ROBERT C.  ...

### Page 899 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 6, Issue 6 [page]

1955 American Mathematical Society. Proceedings of the American Mathematical Society
JAMES A projection in a Banach space is a continuous linear mapping P of the space into itself which is such that P?=P.  ...  Curtiss, “Monte Carlo” methods for the iteration of linear operators, Journal of Mathematics and Physics vol. 32 (1954) pp. 209-232. PRINCETON UNIVERSITY PROJECTIONS IN THE SPACE (m)! ROBERT C.  ...

### Norm One Projections in Banach Spaces [article]

Beata Randrianantoanina
2001 arXiv   pre-print
This is the survey of results about norm one projections and 1-complemented subspaces in K\"othe function spaces and Banach sequence spaces.  ...  [FaHu78, Kin84] Let X be a real Banach space and M a complemented linear subspace of X. Let P be a linear projection from X onto M.  ...  Here by a projection we mean a bounded linear operator P satisfying P 2 = P , and by a complemented subspace we mean a range of a bounded linear projection P .  ...

### Projections in the space of bounded linear operators

David Arterburn, Robert Whitley
1965 Pacific Journal of Mathematics
We recall that a subspace of a Banach space X is said to be complemented (in X) if there is a continuous linear projection of X onto that subspace" In  it is shown that for X and Y certain Banach spaces  ...  Thorp has shown that for X and Y certain Banach spaces of sequences there is no continuous linear projection of the bounded linear operators from X to Y onto the compact linear operators from X to Y.  ...  Suppose that X x and Y 1 are complemented subspaces PROJECTIONS IN THE SPACE OF BOUNDED LINEAR OPERATORS 741 of, respectively, X and Y.  ...

### Extension properties of Banach spaces

Andrew Sobczyk
1962 Bulletin of the American Mathematical Society
properties concerning projections and extensions in Banach spaces are of interest.  ...  In case L and M are Hilbert spaces, g is isometric, and p and g are orthogonal projections, the relative inverse b coincides with the adjoint a* of a.  ...  Since B has the extension property, g has a continuous extension to (m), and Bi is therefore complemented in (m), also in every C(E) since (m) has the projection property.  ...

### Structural Projections on JBW*-Triples

C. Martin Edwards, Gottfried T. Rüttimann
1996 Journal of the London Mathematical Society
A linear projection R on a Jordan*-triple A is said to be structural provided that, for all elements a, b and c in A, the equality {Rab Re} = R{a Rbc} holds.  ...  It is shown that a subtriple of a JBW*-triple is complemented if and only if it is the range of a structural projection.  ...  The authors are grateful for the support for their research has received from the United Kingdom Science and Engineering Research Council and the Schweizerische Nationalfonds/Fonds national suisse.  ...

### On Extensions of Bilinear Maps [article]

C.S. Kubrusly
2021 arXiv   pre-print
It gives a necessary and sufficient condition for extending a bounded bilinear map on the Cartesian product of subspaces of Banach spaces.  ...  Applications concerning projective tensor products are also investigated.  ...  (o) M and N are complemented in X and Y with M = R(E) and N = R(P ) for projections E ∈ B[X , X ] and P ∈ B[Y, Y] such that E = P = 1. (a) M ⊗ ∧ N is a linear manifold of X ⊗ ∧ Y.  ...

### Page 10041 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews
The authors study the case of spine spaces of linear complements. They investigate and compare the geometry of affine spine spaces and affine spaces of linear complements.  ...  For convenience, instead of an affine space of linear complements itself they use its repre- sentation as the space of appropriate linear maps.  ...

### The non-conjugacy of certain algebras of operators

Julien Hennefeld
1972 Pacific Journal of Mathematics
Let E be a Banach space and B{E) be the space of all bounded linear operators on E.  ...  In this paper it is proved that if E has an unconditional basis and is not isomorphic to a conjugate space, then B(E) is not isomorphic to a conjugate space. An even stronger result is proved.  ...  For X and E Banach spaces let B(X, E) denote the space of all bounded linear operators from X into E. THEOREM Then & is a projection onto a subspace isomorphic to r^( M).  ...

### A sufficient condition for the sum of complemented subspaces to be complemented

I.S. Feshchenko
2019 Natsional'na Akademiya Nauk Ukrainy. Dopovidi: naukovyi zhurnal
Under this condition, the formula for a projection onto the sum is given. The condition is sharp (in a certain sense).  ...  Samoilenko We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be complemented.  ...  M is said to be complemented in X if there exists a continuous linear projection onto M , i.e., a continuous linear operator → : P X X such that ∈ Px M for all ∈ x X and = Px x for ∈ x M .  ...
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