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Journal articles on the topic 'Regularisation in Banach spaces'

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Published: 7 July 2024

Last updated: 7 July 2024

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1

Simons, S. "Regularisations of convex functions and slicewise suprema." Bulletin of the Australian Mathematical Society 50, no. 3 (December 1994): 481–99. http://dx.doi.org/10.1017/s0004972700013599.

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For a number of years, there has been interest in the regularisation of a given proper convex lower semicontinuous function on a Banach space, defined to be the episum (=inf-convolution) of the function with a scalar multiple of the norm. There is an obvious geometric way of characterising this regularisation as the lower envelope of cones lying above the graph of the original function. In this paper, we consider the more interesting problem of characterising the regularisation in terms of approximations from below, expressing the regularisation as the upper envelope of certain subtangents to the graph of the original function. We shall show that such an approximation is sometimes (but not always) valid. Further, we shall give an extension of the whole procedure in which the scalar multiple of the norm is replaced by a more general sublinear functional. As a by-product of our analysis, we are led to the consideration of two senses stronger than the pointwise sense in which a function on a Banach space can be expressed as the upper envelope of a family of functions. These new senses of suprema lead to some questions in Banach space theorey.

2

Werner, Dirk. "Indecomposable Banach spaces." Acta et Commentationes Universitatis Tartuensis de Mathematica 5 (December 31, 2001): 89–105. http://dx.doi.org/10.12697/acutm.2001.05.08.

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This paper aims at describing Tim Gowers' contributions to Banach space theory that earned him the Fields medal in 1998. In particular, the construction of the Gowers-Maurey space, a Banach space not containing an unconditional basic sequence, is sketched as is the Gowers dichotomy theorem that led to the solution of the homogeneous Banach space problem. Moreover, Gowers' counterexamples to the hyperplane problem and the Schröder-Bernstein problem are discussed. The paper is an extended version of a talk given at Freie Universität Berlin in December 1999; hence the reference to the next millennium at the very end actually appeals to the present millennium. It should be accessible to anyone with a basic knowledge of functional analysis and of German.

3

Kusraev, A. G. "Banach-Kantorovich spaces." Siberian Mathematical Journal 26, no. 2 (1985): 254–59. http://dx.doi.org/10.1007/bf00968770.

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4

Oikhberg, T., and E. Spinu. "Subprojective Banach spaces." Journal of Mathematical Analysis and Applications 424, no. 1 (April 2015): 613–35. http://dx.doi.org/10.1016/j.jmaa.2014.11.008.

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5

González, Manuel, and Javier Pello. "Superprojective Banach spaces." Journal of Mathematical Analysis and Applications 437, no. 2 (May 2016): 1140–51. http://dx.doi.org/10.1016/j.jmaa.2016.01.033.

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6

Qiu, Jing Hui, and Kelly McKennon. "Banach-Mackey spaces." International Journal of Mathematics and Mathematical Sciences 14, no. 2 (1991): 215–19. http://dx.doi.org/10.1155/s0161171291000224.

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In recent publications the concepts of fast completeness and local barreledness have been shown to be related to the property of all weak-*bounded subsets of the dual (of a locally convex space) being strongly bounded. In this paper we clarify those relationships, as well as giving several different characterizations of this property.

7

Dineen, Seán, and Michael Mackey. "Confined Banach spaces." Archiv der Mathematik 87, no. 3 (September 2006): 227–32. http://dx.doi.org/10.1007/s00013-006-1693-y.

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8

Ferenczi, Valentin, and Christian Rosendal. "Ergodic Banach spaces." Advances in Mathematics 195, no. 1 (August 2005): 259–82. http://dx.doi.org/10.1016/j.aim.2004.08.008.

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9

Bastero, Jesús. "Embedding unconditional stable banach spaces into symmetric stable banach spaces." Israel Journal of Mathematics 53, no. 3 (December 1986): 373–80. http://dx.doi.org/10.1007/bf02786569.

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10

SHEKHAR, CHANDER, TARA ., and GHANSHYAM SINGH RATHORE. "RETRO K-BANACH FRAMES IN BANACH SPACES." Poincare Journal of Analysis and Applications 05, no. 2.1 (December 30, 2018): 65–75. http://dx.doi.org/10.46753/pjaa.2018.v05i02(i).003.

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11

SHARMA, SHALU. "ON BI-BANACH FRAMES IN BANACH SPACES." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 02 (March 2014): 1450015. http://dx.doi.org/10.1142/s0219691314500155.

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Bi-Banach frames in Banach spaces have been defined and studied. A necessary and sufficient condition under which a Banach space has a Bi-Banach frame has been given. Finally, Pseudo exact retro Banach frames have been defined and studied.

12

Hõim, Terje, and David A. Robbins. "Banach-Stone theorems for Banach bundles." Acta et Commentationes Universitatis Tartuensis de Mathematica 9 (December 31, 2005): 65–76. http://dx.doi.org/10.12697/acutm.2005.09.08.

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We prove two Banach-Stone type theorems for section spaces of real Banach bundles. The first theorem assumes that the duals of all fibers are strictly convex, and the second considers disjointness-preserving operators. In each case, the result generalizes the corresponding Banach-Stone theorem for spaces of continuous vector-valued functions.

13

Tan, Dongni, and Xujian Huang. "The wigner property for CL-spaces and finite-dimensional polyhedral Banach spaces." Proceedings of the Edinburgh Mathematical Society 64, no. 2 (April 30, 2021): 183–99. http://dx.doi.org/10.1017/s0013091521000079.

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AbstractWe 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.

14

JAIN, P. K., S. K. KAUSHIK, and NISHA GUPTA. "ON FRAME SYSTEMS IN BANACH SPACES." International Journal of Wavelets, Multiresolution and Information Processing 07, no. 01 (January 2009): 1–7. http://dx.doi.org/10.1142/s021969130900274x.

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Banach frame systems in Banach spaces have been defined and studied. A sufficient condition under which a Banach space, having a Banach frame, has a Banach frame system has been given. Also, it has been proved that a Banach space E is separable if and only if E* has a Banach frame ({φn},T) with each φn weak*-continuous. Finally, a necessary and sufficient condition for a Banach Bessel sequence to be a Banach frame has been given.

15

Godefroy, G., and N. J. Kalton. "Lipschitz-free Banach spaces." Studia Mathematica 159, no. 1 (2003): 121–41. http://dx.doi.org/10.4064/sm159-1-6.

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16

Wojtowicz, Marek. "Finitely Nonreflexive Banach Spaces." Proceedings of the American Mathematical Society 106, no. 4 (August 1989): 961. http://dx.doi.org/10.2307/2047280.

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17

Lin, Bor-Luh, and T. S. S. R. K. Rao. "Multismoothness in Banach Spaces." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–12. http://dx.doi.org/10.1155/2007/52382.

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In this paper, motivated by the results published by R. Khalil and A. Saleh in 2005, we study the notion ofk-smooth points and the notion ofk-smoothness, which are dual to the notion ofk-rotundity. Generalizing these notions and combining smoothness with the recently introduced notion of unitary, we study classes of Banach spaces for which the vector space, spanned by the state space corresponding to a unit vector, is a closed set.

18

Casazza, P. G., and M. C. Lammers. "Genus $n$ Banach spaces." Illinois Journal of Mathematics 43, no. 2 (June 1999): 307–23. http://dx.doi.org/10.1215/ijm/1255985217.

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19

Lindenstrauss, Joram. "BANACH SPACES FOR ANALYSTS." Bulletin of the London Mathematical Society 24, no. 6 (November 1992): 620–22. http://dx.doi.org/10.1112/blms/24.6.620.

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20

Jayne, J. E., I. Namioka, and C. A. Rogers. "σ‐fragmentable Banach spaces." Mathematika 39, no. 1 (June 1992): 161–88. http://dx.doi.org/10.1112/s0025579300006926.

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21

Jayne, J. E., I. Namioka, and C. A. Rogers. "σ‐fragmentable Banach spaces." Mathematika 39, no. 2 (December 1992): 197–215. http://dx.doi.org/10.1112/s0025579300014935.

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22

Gilles, Godefroy. "Lipschitz approximable Banach spaces." Commentationes Mathematicae Universitatis Carolinae 61, no. 2 (November 5, 2020): 187–93. http://dx.doi.org/10.14712/1213-7243.2020.021.

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23

W{ójtowicz, Marek. "Finitely nonreflexive Banach spaces." Proceedings of the American Mathematical Society 106, no. 4 (April 1, 1989): 961. http://dx.doi.org/10.1090/s0002-9939-1989-0949882-0.

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24

Moreno, Yolanda, and Anatolij Plichko. "On automorphic Banach spaces." Israel Journal of Mathematics 169, no. 1 (November 22, 2008): 29–45. http://dx.doi.org/10.1007/s11856-009-0002-4.

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25

Bandyopadhyay, Pradipta, Yongjin Li, Bor-Luh Lin, and Darapaneni Narayana. "Proximinality in Banach spaces." Journal of Mathematical Analysis and Applications 341, no. 1 (May 2008): 309–17. http://dx.doi.org/10.1016/j.jmaa.2007.10.024.

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26

Rosenthal, Haskell. "Weak∗-Polish Banach spaces." Journal of Functional Analysis 76, no. 2 (February 1988): 267–316. http://dx.doi.org/10.1016/0022-1236(88)90039-0.

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27

Klimek, Slawomir, and Andrzej Lesniewski. "Pfaffians on Banach spaces." Journal of Functional Analysis 102, no. 2 (December 1991): 314–30. http://dx.doi.org/10.1016/0022-1236(91)90124-n.

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28

Abrahamsen, Trond A., Johann Langemets, and Vegard Lima. "Almost square Banach spaces." Journal of Mathematical Analysis and Applications 434, no. 2 (February 2016): 1549–65. http://dx.doi.org/10.1016/j.jmaa.2015.09.060.

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29

Raja, M. "Super WCG Banach spaces." Journal of Mathematical Analysis and Applications 439, no. 1 (July 2016): 183–96. http://dx.doi.org/10.1016/j.jmaa.2016.02.057.

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30

Aron, Richard M., and Seán Dineen. "$Q$-Reflexive Banach Spaces." Rocky Mountain Journal of Mathematics 27, no. 4 (December 1997): 1009–25. http://dx.doi.org/10.1216/rmjm/1181071856.

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31

James, R. C. "Some Interesting Banach Spaces." Rocky Mountain Journal of Mathematics 23, no. 3 (September 1993): 911–37. http://dx.doi.org/10.1216/rmjm/1181072532.

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32

Bandyopadhyay, Pradipta, Krzysztof Jarosz, and T. S. S. R. K. Rao. "Unitaries in Banach spaces." Illinois Journal of Mathematics 48, no. 1 (January 2004): 339–51. http://dx.doi.org/10.1215/ijm/1258136187.

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33

Becerra Guerrero, Julio, Antonio Moreno Galindo, and Ángel Rodríguez Palacios. "Absolute-valuable Banach spaces." Illinois Journal of Mathematics 49, no. 1 (January 2005): 121–38. http://dx.doi.org/10.1215/ijm/1258138309.

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34

Ferenczi, V. "Lipschitz Homogeneous Banach Spaces." Quarterly Journal of Mathematics 54, no. 4 (December 1, 2003): 415–19. http://dx.doi.org/10.1093/qmath/hag022.

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35

Bhakta, P. C., and Sumitra Roychaudhuri. "Optimization in Banach spaces." Journal of Mathematical Analysis and Applications 134, no. 2 (September 1988): 460–70. http://dx.doi.org/10.1016/0022-247x(88)90035-2.

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36

Rosendal, Christian. "α-Minimal Banach spaces." Journal of Functional Analysis 262, no. 8 (April 2012): 3638–64. http://dx.doi.org/10.1016/j.jfa.2012.02.001.

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37

Ostrovskii, M. I. "Separably injective Banach spaces." Functional Analysis and Its Applications 20, no. 2 (1986): 154–55. http://dx.doi.org/10.1007/bf01077281.

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38

Johanis, Michal. "Locally flat Banach spaces." Czechoslovak Mathematical Journal 59, no. 1 (March 2009): 273–84. http://dx.doi.org/10.1007/s10587-009-0019-1.

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39

Terekhin, P. A. "Frames in Banach spaces." Functional Analysis and Its Applications 44, no. 3 (September 2010): 199–208. http://dx.doi.org/10.1007/s10688-010-0024-z.

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40

Dinculeanu, Nicolae, and Muthu Muthiah. "Bimeasures in Banach spaces." Annali di Matematica Pura ed Applicata 178, no. 1 (December 2000): 339–92. http://dx.doi.org/10.1007/bf02505903.

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41

Argyros, S., S. Negrepontis, and Th Zachariades. "Weakly stable Banach spaces." Israel Journal of Mathematics 57, no. 1 (February 1987): 68–88. http://dx.doi.org/10.1007/bf02769461.

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42

Banakh, Taras, and Joanna Garbulińska-Wȩgrzyn. "Universal decomposed Banach spaces." Banach Journal of Mathematical Analysis 14, no. 2 (January 1, 2020): 470–86. http://dx.doi.org/10.1007/s43037-019-00003-7.

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AbstractLet $${\mathcal {B}}$$B be a class of finite-dimensional Banach spaces. A $${\mathcal {B}}$$B-decomposed Banach space is a Banach space X endowed with a family $${\mathcal {B}}_X\subset {\mathcal {B}}$$BX⊂B of subspaces of X such that each $$x\in X$$x∈X can be uniquely written as the sum of an unconditionally convergent series $$\sum _{B\in {\mathcal {B}}_X}x_B$$∑B∈BXxB for some $$(x_B)_{B\in {\mathcal {B}}_X}\in \prod _{B\in {\mathcal {B}}_X}B$$(xB)B∈BX∈∏B∈BXB. For every $$B\in {\mathcal {B}}_X$$B∈BX let $$\mathrm {pr}_B:X\rightarrow B$$prB:X→B denote the coordinate projection. Let $$C\subset [-1,1]$$C⊂[-1,1] be a closed convex set with $$C\cdot C\subset C$$C·C⊂C. The C-decomposition constant $$K_C$$KC of a $${\mathcal {B}}$$B-decomposed Banach space $$(X,{\mathcal {B}}_X)$$(X,BX) is the smallest number $$K_C$$KC such that for every function $$\alpha :{\mathcal {F}}\rightarrow C$$α:F→C from a finite subset $${\mathcal {F}}\subset {\mathcal {B}}_X$$F⊂BX the operator $$T_\alpha =\sum _{B\in {\mathcal {F}}}\alpha (B)\cdot \mathrm {pr}_B$$Tα=∑B∈Fα(B)·prB has norm $$\Vert T_\alpha \Vert \le K_C$$‖Tα‖≤KC. By $$\varvec{{\mathcal {B}}}_C$$BC we denote the class of $${\mathcal {B}}$$B-decomposed Banach spaces with C-decomposition constant $$K_C\le 1$$KC≤1. Using the technique of Fraïssé theory, we construct a rational $${\mathcal {B}}$$B-decomposed Banach space $$\mathbb {U}_C\in \varvec{{\mathcal {B}}}_C$$UC∈BC which contains an almost isometric copy of each $${\mathcal {B}}$$B-decomposed Banach space $$X\in \varvec{{\mathcal {B}}}_C$$X∈BC. If $${\mathcal {B}}$$B is the class of all 1-dimensional (resp. finite-dimensional) Banach spaces, then $$\mathbb {U}_{C}$$UC is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional (f.d.) basis, constructed by Pełczyński (and Wojtaszczyk).

43

Agud, Lucia, Jose Manuel Calabuig, Maria Aranzazu Juan, and Enrique A. Sánchez Pérez. "Banach Lattice Structures and Concavifications in Banach Spaces." Mathematics 8, no. 1 (January 14, 2020): 127. http://dx.doi.org/10.3390/math8010127.

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Let ( Ω , Σ , μ ) be a finite measure space and consider a Banach function space Y ( μ ) . We say that a Banach space E is representable by Y ( μ ) if there is a continuous bijection I : Y ( μ ) → E . In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.

44

Cho, Chong-Man. "OPERATORS FROM CERTAIN BANACH SPACES TO BANACH SPACES OF COTYPE q ≥ 2." Communications of the Korean Mathematical Society 17, no. 1 (January 1, 2002): 53–56. http://dx.doi.org/10.4134/ckms.2002.17.1.053.

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45

Schlumprecht, Th. "On Zippin's Embedding Theorem of Banach spaces into Banach spaces with bases." Advances in Mathematics 274 (April 2015): 833–80. http://dx.doi.org/10.1016/j.aim.2015.02.004.

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46

Soybaş, Danyal. "The () Property in Banach Spaces." Abstract and Applied Analysis 2012 (2012): 1–7. http://dx.doi.org/10.1155/2012/754531.

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A Banach space is said to have (D) property if every bounded linear operator is weakly compact for every Banach space whose dual does not contain an isomorphic copy of . Studying this property in connection with other geometric properties, we show that every Banach space whose dual has (V∗) property of Pełczyński (and hence every Banach space with (V) property) has (D) property. We show that the space of real functions, which are integrable with respect to a measure with values in a Banach space , has (D) property. We give some other results concerning Banach spaces with (D) property.

47

Lima, Vegard. "Approximation properties for dual spaces." MATHEMATICA SCANDINAVICA 93, no. 2 (December 1, 2003): 297. http://dx.doi.org/10.7146/math.scand.a-14425.

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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 $X$.

48

Cui, Yunan, Henryk Hudzik, and Ryszard Płuciennik. "Banach-Saks property in some Banach sequence spaces." Annales Polonici Mathematici 65, no. 2 (1997): 193–202. http://dx.doi.org/10.4064/ap-65-2-193-202.

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49

Brech, C., and C. Piña. "Banach-Stone-like results for combinatorial Banach spaces." Annals of Pure and Applied Logic 172, no. 8 (August 2021): 102989. http://dx.doi.org/10.1016/j.apal.2021.102989.

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50

Bernal-González, L., J. Fernández-Sánchez, M. E. Martínez-Gómez, and J. B. Seoane-Sepúlveda. "Banach spaces and Banach lattices of singular functions." Studia Mathematica 260, no. 2 (2021): 167–93. http://dx.doi.org/10.4064/sm200419-7-9.

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