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

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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The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091521000250 ◽

2021 ◽

pp. 1-17

Author(s):

Dongni Tan ◽

Xujian Huang

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Phase Function ◽

Linear Isometry ◽

Basic Properties ◽

Finite Dimensional ◽

Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

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How low can vacuum energy go when your fields are finite-dimensional?

International Journal of Modern Physics D ◽

10.1142/s0218271819440061 ◽

2019 ◽

Vol 28 (14) ◽

pp. 1944006

Author(s):

ChunJun Cao ◽

Aidan Chatwin-Davies ◽

Ashmeet Singh

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Degrees Of Freedom ◽

Vacuum Energy ◽

Quantum Field ◽

Locally Finite ◽

Finite Dimensional ◽

Finite Dimensionality ◽

Klein Gordon ◽

Dimensional Version

According to the holographic bound, there is only a finite density of degrees of freedom in space when gravity is taken into account. Conventional quantum field theory does not conform to this bound, since in this framework, infinitely many degrees of freedom may be localized to any given region of space. In this paper, we explore the viewpoint that quantum field theory may emerge from an underlying theory that is locally finite-dimensional, and we construct a locally finite-dimensional version of a Klein–Gordon scalar field using generalized Clifford algebras. Demanding that the finite-dimensional field operators obey a suitable version of the canonical commutation relations makes this construction essentially unique. We then find that enforcing local finite dimensionality in a holographically consistent way leads to a huge suppression of the quantum contribution to vacuum energy, to the point that the theoretical prediction becomes plausibly consistent with observations.

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In what spaces is every closed normal cone regular?

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150000938x ◽

1970 ◽

Vol 17 (2) ◽

pp. 121-125 ◽

Cited By ~ 8

Author(s):

C. W. McArthur

Keyword(s):

Banach Space ◽

Convex Space ◽

Unconditional Basis ◽

Normal Cone ◽

Closed Subspace ◽

Locally Convex Space ◽

Regular Part ◽

Fréchet Space ◽

Sequential Completeness ◽

Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

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The measure problem in no-collapse (many worlds) quantum mechanics

International Journal of Modern Physics D ◽

10.1142/s0218271817300087 ◽

2017 ◽

Vol 26 (03) ◽

pp. 1730008 ◽

Cited By ~ 2

Author(s):

Stephen D. H. Hsu

Keyword(s):

Wave Function ◽

Quantum Mechanics ◽

Probability Measure ◽

Ab Initio ◽

Subjective Probability ◽

State Vector ◽

A Priori ◽

Born Rule ◽

Many Worlds ◽

Measure Problem

We explain the measure problem (cf. origin of the Born probability rule) in no-collapse quantum mechanics. Everett defined maverick branches of the state vector as those on which the usual Born probability rule fails to hold — these branches exhibit highly improbable behaviors, including possibly the breakdown of decoherence or even the absence of an emergent semi-classical reality. Derivations of the Born rule which originate in decision theory or subjective probability (i.e. the reasoning of individual observers) do not resolve this problem, because they are circular: they assume, a priori, that the observer occupies a non-maverick branch. An ab initio probability measure is sometimes assumed to explain why we do not occupy a maverick branch. This measure is constrained by, e.g. Gleason’s theorem or envariance to be the usual Hilbert measure. However, this ab initio measure ultimately governs the allocation of a self or a consciousness to a particular branch of the wave function, and hence invokes primitives which lie beyond the Everett wave function and beyond what we usually think of as physics. The significance of this leap has been largely overlooked, but requires serious scrutiny.

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Best Simultaneous Approximation in Orlicz Spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2007/68017 ◽

2007 ◽

Vol 2007 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

M. Khandaqji ◽

Sh. Al-Sharif

Keyword(s):

Banach Space ◽

Simultaneous Approximation ◽

Closed Subspace ◽

Orlicz Spaces ◽

Unit Interval ◽

Luxemburg Norm ◽

Integrable Functions ◽

Best Simultaneous Approximation ◽

Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

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On the convergence of greedy algorithms for initial segments of the Haar basis

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004109990478 ◽

2010 ◽

Vol 148 (3) ◽

pp. 519-529 ◽

Cited By ~ 1

Author(s):

S. J. DILWORTH ◽

E. ODELL ◽

TH. SCHLUMPRECHT ◽

ANDRÁS ZSÁK

Keyword(s):

Banach Space ◽

Greedy Algorithm ◽

General Result ◽

Initial Segment ◽

Greedy Algorithms ◽

Dimensional Banach Space ◽

Haar Basis ◽

Finite Dimensional ◽

Finite Dimensional Banach Space ◽

Number Of Iterations

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

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Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

CAUCHY ◽

10.18860/ca.v4i4.4100 ◽

2017 ◽

Vol 4 (4) ◽

pp. 167

Author(s):

Minanur Rohman

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Spaces ◽

Lorentz Space ◽

Bounded Linear Operator ◽

Closed Subspace ◽

Reflexive Banach Space ◽

Reflexive Banach Spaces ◽

Bounded Linear ◽

Star Topology

In this paper, we will discuss some applications of almost surjective epsilon-isometry mapping, one of them is in Lorentz space ( L_(p,q)-space). Furthermore, using some classical theorems of w star-topology and concept of closed subspace -complemented, for every almost surjective epsilon-isometry mapping f : X to Y, where Y is a reflexive Banach space, then there exists a bounded linear operator T : Y to X with such that for every x in X.

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Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽

10.3390/math9182346 ◽

2021 ◽

Vol 9 (18) ◽

pp. 2346

Author(s):

Almudena Campos-Jiménez ◽

Francisco Javier García-Pacheco

Keyword(s):

Banach Space ◽

Unit Ball ◽

Banach Spaces ◽

Unit Sphere ◽

Geometric Property ◽

The Other ◽

Geometric Invariants ◽

Finite Dimensional ◽

Surjective Isometry ◽

Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

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Homological characterizations of the approximation property for Banach spaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500008776 ◽

1992 ◽

Vol 34 (2) ◽

pp. 229-239 ◽

Cited By ~ 8

Author(s):

Yu. V. Selivanov

Keyword(s):

Banach Space ◽

Banach Algebra ◽

Banach Spaces ◽

Cohomology Group ◽

Approximation Property ◽

Three Dimensional ◽

Nuclear Operators ◽

Finite Dimensionality

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)).

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A characterization of M-Summands

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100048817 ◽

1974 ◽

Vol 76 (1) ◽

pp. 157-159 ◽

Cited By ~ 1

Author(s):

Richard Evans

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Closed Subspace ◽

Structure Theory ◽

Linear Projection ◽

Image Position

In the structure theory of Banach spaces as developed in (1), an important role is played by subspaces which are the ranges of projections having norm properties akin to those of the classical Banach spaces. A linear projection e on a Banach space V is called an M-projection ifand an L-projection if, insteadA closed subspace J of V is called an M-Summand if it is the range of an M-projection and an M-Ideal if J0 is the range of an L-projection in V′. Every M-Summand is an M-Ideal but the reverse is false.

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