scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

2022 ◽  
Author(s):  
◽  
Long Qian

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>


2020 ◽  
Vol 18 (1) ◽  
pp. 1698-1708
Author(s):  
Ju Myung Kim

Abstract For a finitely generated tensor norm α \alpha , we investigate the α \alpha -approximation property ( α \alpha -AP) and the bounded α \alpha -approximation property (bounded α \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ \lambda -bounded α p , q {\alpha }_{p,q} -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ \lambda -bounded g p {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the λ \lambda -bounded g p {g}_{p} -AP, then X has the λ \lambda -bounded w p {w}_{p} -AP.


2003 ◽  
Vol 93 (2) ◽  
pp. 297 ◽  
Author(s):  
Vegard Lima

We prove that a Banach space $X$ has the metric approximation property if and only if $\mathcal F(Y,X)$ is an ideal in $\mathcal L(Y,X^{**})$ for all Banach spaces $Y$. Furthermore, $X^*$ has the metric approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $\phi : X^* \rightarrow X^{***}$ there exists a Hahn-Banach extension operator $\Phi : {\mathcal F(Y,X)}^* \rightarrow {\mathcal L(Y,X^{**})}^*$ such that $\Phi(x^* \otimes y^{**}) = (\phi x^*) \otimes y^{**}$ for all $x^* \in X^*$ and all $y^{**} \in Y^{**}$. We also prove that $X^*$ has the approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $\phi : X^* \rightarrow X^{***}$ there exists a Hahn-Banach extension operator $\Phi : {\mathcal F(Y,X)}^* \rightarrow {\mathcal W(Y,X^{**})}^*$ such that $\Phi(x^* \otimes y^{**}) = (\phi x^*) \otimes y^{**}$ for all $x^* \in X^*$ and all $y^{**} \in Y^{**}$, which in turn is equivalent to $\mathcal F(Y,\hat{X})$ being an ideal in $\mathcal W(Y,\hat{X}^{**})$ for all Banach spaces $Y$ and all equivalent renormings $\hat{X}$ of $X$.


Author(s):  
Eve Oja ◽  
Indrek Zolk

Let a, c ≥ 0 and let B be a compact set of scalars. We show that if X is a Banach space such that the canonical projection π from X*** onto X* satisfies the inequalityand 1 ≤ λ < max |B| + c, then every λ-commuting bounded compact approximation of the identity of X is shrinking. This generalizes a theorem by Godefroy and Saphar from 1988. As an application, we show that under the conditions described above both X and X* have the metric compact approximation property (MCAP). Relying on the Willis construction, we show that the commuting MCAP does not imply the approximation property.


2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.


1992 ◽  
Vol 34 (2) ◽  
pp. 229-239 ◽  
Author(s):  
Yu. V. Selivanov

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:(i) biprojectivity of N(E);(ii) biflatness of N(E);(iii) homological finite-dimensionality of N(E);(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).


1991 ◽  
Vol 44 (1) ◽  
pp. 75-90 ◽  
Author(s):  
David P. Blecher

AbstractTogether with Vern Paulsen we were able to show that the elementary theory of tensor norms of Banach spaces carries over to operator spaces. We suggested that the Grothendieck tensor norm program, which was of course enormously important in the development of Banach space theory, be carried out for operator spaces. Some of this has been done by the authors mentioned above, and by Effros and Ruan. We give alternative developments of some of this work, and otherwise continue the tensor norm program.


1993 ◽  
Vol 03 (04) ◽  
pp. 313-320 ◽  
Author(s):  
PHILIP D. MACKENZIE

In the postal model of message passing systems, the actual communication network between processors is abstracted by a single communication latency factor, which measures the inverse ratio of the time it takes for a processor to send a message and the time that passes until the recipient receives the message. In this paper we examine the problem of broadcasting multiple messages in an order-preserving fashion in the postal model. We prove lower bounds for all parameter ranges and show that these lower bounds are within a factor of seven of the best upper bounds. In some cases, our lower bounds show significant asymptotic improvements over the previous best lower bounds.


1949 ◽  
Vol 14 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Frederic B. Fitch

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.


2016 ◽  
Vol 24 (4) ◽  
pp. 719-744 ◽  
Author(s):  
Maxim Buzdalov ◽  
Benjamin Doerr ◽  
Mikhail Kever

We analyze the unrestricted black-box complexity of the Jump function classes for different jump sizes. For upper bounds, we present three algorithms for small, medium, and extreme jump sizes. We prove a matrix lower bound theorem which is capable of giving better lower bounds than the classic information theory approach. Using this theorem, we prove lower bounds that almost match the upper bounds. For the case of extreme jump functions, which apart from the optimum reveal only the middle fitness value(s), we use an additional lower bound argument to show that any black-box algorithm does not gain significant insight about the problem instance from the first [Formula: see text] fitness evaluations. This, together with our upper bound, shows that the black-box complexity of extreme jump functions is [Formula: see text].


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