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Journal articles on the topic 'Cotype and type of Banach spaces'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Mastylo, Mieczyslaw. "Type and cotype of some Banach spaces." International Journal of Mathematics and Mathematical Sciences 15, no. 2 (1992): 235–40. http://dx.doi.org/10.1155/s0161171292000309.

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Type and cotype are computed for Banach spaces generated by some positive sublinear operators and Banach function spaces. Applications of the results yield that under certain assumptions Clarkson's inequalities hold in these spaces.

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2

Basallote, Manuela, Manuel D. Contreras, and Santiago Díaz-Madrigal. "Uniformly convexifying operators in classical Banach spaces." Bulletin of the Australian Mathematical Society 59, no. 2 (April 1999): 225–36. http://dx.doi.org/10.1017/s0004972700032846.

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We obtain a new characterisation of finite representability of operators and present new results about uniformly convexifying, Rademacher cotype and Rademacher type operators on some classical Banach spaces, including JB* -triples and spaces of analytic functions.

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3

Pietsch, Albrecht. "Type and cotype numbers of operators on Banach spaces." Studia Mathematica 96, no. 1 (1990): 21–37. http://dx.doi.org/10.4064/sm-96-1-21-37.

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4

Kato, Mikio, and Yasuji Takahashi. "Type, Cotype Constants and Clarkson's Inequalities for Banach Spaces." Mathematische Nachrichten 186, no. 1 (1997): 187–96. http://dx.doi.org/10.1002/mana.3211860111.

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5

Cuesta, Javier. "Type and Cotype Constants and the Linear Stability of Wigner’s Symmetry Theorem." Symmetry 11, no. 9 (September 3, 2019): 1107. http://dx.doi.org/10.3390/sym11091107.

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We study the relation between almost-symmetries and the geometry of Banach spaces. We show that any almost-linear extension of a transformation that preserves transition probabilities up to an additive error admits an approximation by a linear map, and the quality of the approximation depends on the type and cotype constants of the involved spaces.

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6

Torrea, José L., and Chao Zhang. "Fractional vector-valued Littlewood–Paley–Stein theory for semigroups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 144, no. 3 (May 16, 2014): 637–67. http://dx.doi.org/10.1017/s0308210511001302.

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We consider the fractional derivative of a general Poisson semigroup. With this fractional derivative, we define the generalized fractional Littlewood–Paley g-function for semigroups acting on Lp-spaces of functions with values in Banach spaces. We give a characterization of the classes of Banach spaces for which the fractional Littlewood–Paley g-function is bounded on Lp-spaces. We show that the class of Banach spaces is independent of the order of derivation and coincides with the classical (Lusin-type/-cotype) case. We also show that the same kind of results exist for the case of the fractional area function and the fractional gλ*-function on ℝn. Finally, we consider the relationship of the almost sure finiteness of the fractional Littlewood–Paley g-function, the area function and the gλ*-function with the Lusin-cotype property of the underlying Banach space. As a byproduct of the techniques developed, one can find some results of independent interest for vector-valued Calderón–Zygmund operators. For example, one can find the following characterization: a Banach space is the unconditional martingale difference if and only if, for some (or, equivalently, for every) p ∈ [1, ∞), dy exists for almost every x ∈ ℝ and every .

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7

Ding, Longyun. "Borel reductibility and Hölder (α) embeddability between Banach spaces." Journal of Symbolic Logic 77, no. 1 (March 2012): 224–44. http://dx.doi.org/10.2178/jsl/1327068700.

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AbstractWe investigate Borel reducibility between equivalence relations E(X; p) = Xℕ/ℓp(X)'s where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between E(Lr; p)'s and E(c0; p)'s for r, p Є [1, +∞).We also answer a problem presented by Kanovei in the affirmative by showing that C(ℝ+)/C0(ℝ+) is Borel bireducible to ℝℕ/c0.

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8

Kamińska, A., L. Maligranda, and L. E. Persson. "Indices, convexity and concavity of Calderón-Lozanovskii spaces." MATHEMATICA SCANDINAVICA 92, no. 1 (March 1, 2003): 141. http://dx.doi.org/10.7146/math.scand.a-14398.

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In this article we discuss lattice convexity and concavity of Calderón-Lozanovskii space $E_\varphi$, generated by a quasi-Banach space $E$ and an increasing Orlicz function $\varphi$. We give estimations of convexity and concavity indices of $E_\varphi$ in terms of Matuszewska-Orlicz indices of $\varphi$ as well as convexity and concavity indices of $E$. In the case when $E_\varphi$ is a rearrangement invariant space we also provide some estimations of its Boyd indices. As corollaries we obtain some necessary and sufficient conditions for normability of $E_\varphi$, and conditions on its nontrivial type and cotype in the case when $E_\varphi$ is a Banach space. We apply these results to Orlicz-Lorentz spaces receiving estimations, and in some cases the exact values of their convexity, concavity and Boyd indices.

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9

Chilin, Vladimir I., Andrei V. Krygin, and Pheodor A. Sukochev. "Local uniform and uniform convexity of non-commutative symmetric spaces of measurable operators." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 2 (March 1992): 355–68. http://dx.doi.org/10.1017/s0305004100075459.

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Let E be a separable symmetric sequence space, and let CE be the unitary matrix space associated with E, i.e. the Banach space of all compact operators x on l2 so that s(x) E, with the norm , where are the s-numbers of x. One of the interesting subjects in the theory of the unitary matrix spaces is the clarification of correlation between the geometric properties of the spaces E and CE. A series of results in this direction related with the notions of type, cotype and uniform convexity of the spaces CE has been already obtained (see 13).

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10

VEOMETT, E., and K. WILDRICK. "SPACES OF SMALL METRIC COTYPE." Journal of Topology and Analysis 02, no. 04 (December 2010): 581–97. http://dx.doi.org/10.1142/s1793525310000422.

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Mendel and Naor's definition of metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz is equivalent to an ultrametric space having infimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov–Hausdorff limits, and use these facts to establish a partial converse of the main result.

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11

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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12

Junge, Marius. "On cotype and summing properties in Banach spaces." Illinois Journal of Mathematics 46, no. 2 (April 2002): 331–56. http://dx.doi.org/10.1215/ijm/1258136197.

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13

Milman, Vitali D., and Gilles Pisier. "Banach spaces with a weak cotype 2 property." Israel Journal of Mathematics 54, no. 2 (June 1986): 139–58. http://dx.doi.org/10.1007/bf02764939.

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14

Mendel, Manor, and Assaf Naor. "Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces." Analysis and Geometry in Metric Spaces 1 (May 28, 2013): 163–99. http://dx.doi.org/10.2478/agms-2013-0003.

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Abstract The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.

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15

Mankiewicz, Piotr, and Nicole Tomczak-Jaegermann. "Structural Properties of Weak Cotype 2 Spaces." Canadian Journal of Mathematics 48, no. 3 (June 1, 1996): 607–24. http://dx.doi.org/10.4153/cjm-1996-032-5.

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AbstractSeveral characterizations of weak cotype 2 and weak Hilbert spaces are given in terms of basis constants and other structural invariants of Banach spaces. For finite-dimensional spaces, characterizations depending on subspaces of fixed proportional dimension are proved.

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16

Zhu, J. "The weak cotype 2 and the Orlicz property of the Lorentz sequence space d(a, 1)." Glasgow Mathematical Journal 34, no. 3 (September 1992): 271–76. http://dx.doi.org/10.1017/s0017089500008831.

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The question “Does a Banach space with a symmetric basis and weak cotype 2 (or Orlicz) property have cotype 2?” is being seriously considered but is still open though the similar question for the r.i. function space on [0, 1] has an affirmative answer. (If X is a r.i. function space on [0, 1] and has weak cotype 2 (or Orlicz) property then it must have cotype 2.) In this note we prove that for Lorentz sequence spaces d(a, 1) they both hold.

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17

Lee, Hun Hee. "Type and cotype of operator spaces." Studia Mathematica 185, no. 3 (2008): 219–47. http://dx.doi.org/10.4064/sm185-3-2.

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18

Pisier, Gilles. "The dual J* of the James space has cotype 2 and the Gordon-Lewis property." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 2 (March 1988): 323–31. http://dx.doi.org/10.1017/s0305004100064902.

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AbstractWe prove the result in the title. More generally we consider the Banach space υp, of all sequences (xn) of scalars such thatwhere the supremum runs over all increasing sequences n1 ≤ n2 ≤ …. We show that is of cotype 2 if p ≽ 2 and of cotype p′, where 1/p′ + 1/p = 1, if p ≤ 2. Similar results are obtained for the analogous function spaces.

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19

Wang, Xiang Chen, and M. Bhaskara Rao. "A note on convergence in Banach spaces of cotype p." Statistics & Probability Letters 10, no. 5 (October 1990): 391–96. http://dx.doi.org/10.1016/0167-7152(90)90019-4.

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20

Giladi, Ohad, Manor Mendel, and Assaf Naor. "Improved bounds in the metric cotype inequality for Banach spaces." Journal of Functional Analysis 260, no. 1 (January 2011): 164–94. http://dx.doi.org/10.1016/j.jfa.2010.08.015.

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21

Rowecka, Ewa. "Ramdom integrals and type and cotype of Banach space." Mathematische Zeitschrift 193, no. 3 (September 1986): 381–91. http://dx.doi.org/10.1007/bf01229805.

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22

Komorowski, Ryszard. "Isometric Characterizations of ℓnp Spaces." Canadian Journal of Mathematics 46, no. 3 (June 1, 1994): 574–85. http://dx.doi.org/10.4153/cjm-1994-030-x.

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AbstractThe paper establishes some characterizations of spaces in terms of p-summing or p-nuclear norms of the identity operator on the given space E.In particular, for an n-dimensional Banach space E and I ≤ p < 2, E is isometric to if and only if πp(E*) ≥ n1/p and E* has cotype p' with the constant one.Furthermore, spaces are characterized by inequalities for p-summing norms of operators related to the John's ellipsoid of maximal volume contained in the unit ball of E.

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23

OSTROVSKA, S., and M. I. OSTROVSKII. "DISTORTION IN THE FINITE DETERMINATION RESULT FOR EMBEDDINGS OF LOCALLY FINITE METRIC SPACES INTO BANACH SPACES." Glasgow Mathematical Journal 61, no. 1 (February 6, 2018): 33–47. http://dx.doi.org/10.1017/s0017089518000022.

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AbstractGiven a Banach spaceXand a real number α ≥ 1, we write: (1)D(X) ≤ α if, for any locally finite metric spaceA, all finite subsets of which admit bilipschitz embeddings intoXwith distortions ≤C, the spaceAitself admits a bilipschitz embedding intoXwith distortion ≤ α ⋅C; (2)D(X) = α+if, for every ϵ > 0, the conditionD(X) ≤ α + ϵ holds, whileD(X) ≤ α does not; (3)D(X) ≤ α+ifD(X) = α+orD(X) ≤ α. It is known thatD(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1)D((⊕n=1∞Xn)p) ≤ 1+for every nested family of finite-dimensional Banach spaces {Xn}n=1∞and every 1 ≤p≤ ∞. (2)D((⊕n=1∞ℓ∞n)p) = 1+for 1 <p< ∞. (3)D(X) ≤ 4+for every Banach spaceXwith no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).

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24

Kamińska, A., L. Maligranda, and L. E. Persson. "Convexity, concavity, type and cotype of Lorentz spaces." Indagationes Mathematicae 9, no. 3 (September 1998): 367–82. http://dx.doi.org/10.1016/s0019-3577(98)80006-2.

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25

Katirtzoglou, Eleni. "Type and Cotype of Musielak–Orlicz Sequence Spaces." Journal of Mathematical Analysis and Applications 226, no. 2 (October 1998): 431–55. http://dx.doi.org/10.1006/jmaa.1998.6089.

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26

Hinrichs, Aicke. "Rademacher and Gaussian averages and Rademacher cotype of operators between Banach spaces." Proceedings of the American Mathematical Society 128, no. 1 (June 21, 1999): 203–13. http://dx.doi.org/10.1090/s0002-9939-99-05012-1.

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27

Blasco, Oscar, and Pablo Gregori. "Type and Cotype in Vector-Valued Nakano Sequence Spaces." Journal of Mathematical Analysis and Applications 264, no. 2 (December 2001): 657–72. http://dx.doi.org/10.1006/jmaa.2001.7691.

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28

Yaroslavtsev, Ivan. "Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces." Communications in Mathematical Physics 379, no. 2 (September 16, 2020): 417–59. http://dx.doi.org/10.1007/s00220-020-03845-7.

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Abstract In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that $$M_0=0$$ M 0 = 0 , we show that the following two-sided inequality holds for all $$1\le p<\infty $$ 1 ≤ p < ∞ : Here $$ \gamma ([\![M]\!]_t) $$ γ ( [ [ M ] ] t ) is the $$L^2$$ L 2 -norm of the unique Gaussian measure on X having $$[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle , \langle M,y^*\rangle ]_t$$ [ [ M ] ] t ( x ∗ , y ∗ ) : = [ ⟨ M , x ∗ ⟩ , ⟨ M , y ∗ ⟩ ] t as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of ($$\star $$ ⋆ ) was proved for UMD Banach functions spaces X. We show that for continuous martingales, ($$\star $$ ⋆ ) holds for all $$0<p<\infty $$ 0 < p < ∞ , and that for purely discontinuous martingales the right-hand side of ($$\star $$ ⋆ ) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, ($$\star $$ ⋆ ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of ($$\star $$ ⋆ ) for arbitrary martingales implies the UMD property for X. As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.

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29

Rozendaal, Jan, and Mark Veraar. "Fourier multiplier theorems on Besov spaces under type and cotype conditions." Banach Journal of Mathematical Analysis 11, no. 4 (October 2017): 713–43. http://dx.doi.org/10.1215/17358787-2017-0011.

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30

Martínez, Teresa, and José L. Torrea. "Boundedness of vector-valued martingale transforms on extreme points and applciations." Journal of the Australian Mathematical Society 76, no. 2 (April 2004): 207–22. http://dx.doi.org/10.1017/s1446788700008909.

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AbstractLet Β1, Β2be a pair of Banach spaces andTbe a vector valued martingale transform (with respect to general filtration) which maps Β1-valued martingales into Β2-valued martingales. Then, the following statements are equivalent:Tis bounded fromintofor somep(or equivalently for everyp) in the range 1 <p< ∞;Tis bounded fromintoBMOB2;Tis bounded fromBMOB1intoBMOB2;Tis bounded frominto. Applications toUMDand martingale cotype properties are given. We also prove that the Hardy spacedefined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if Β has the Radon-Nikodým property.

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31

Abraham, Paul. "Saeki's Improvement of the Vitali-Hahn-Saks-Nikodym Theorem Holds Precisely for Banach Spaces having Cotype." Proceedings of the American Mathematical Society 116, no. 1 (September 1992): 171. http://dx.doi.org/10.2307/2159310.

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32

Abraham, Paul. "Saeki’s improvement of the Vitali-Hahn-Saks-Nikodým theorem holds precisely for Banach spaces having cotype." Proceedings of the American Mathematical Society 116, no. 1 (January 1, 1992): 171. http://dx.doi.org/10.1090/s0002-9939-1992-1095219-9.

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33

Nikolova, L. Y., L. E. Persson, and T. Zachariades. "On Clarkson's inequality, type and cotype for the Edmunds-Triebel logarithmic spaces." Archiv der Mathematik 80, no. 2 (April 1, 2003): 165–76. http://dx.doi.org/10.1007/s00013-003-0451-7.

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34

Eryilmaz, İlker. "Sobolev type spaces based on Lorentz-Karamata spaces." Filomat 30, no. 11 (2016): 3023–32. http://dx.doi.org/10.2298/fil1611023e.

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In this paper, firstly Lorentz-Karamata-Sobolev spaces Wk,(p,q,b) (Rn) of integer order are introduced and some of their important properties are emphasized. Also, Banach spaces Ak,L(p,q,b)(Rn) = L1(Rn)? Wk,L(p,q,b)(Rn) (Lorentz-Karamata-Sobolev algebras) are studied. Using a result of H.C.Wang, it is showed that Banach convolution algebras AkL(p,q,b)(Rn) don?t have weak factorization and the multiplier algebra of Ak,L(p,q,b)(Rn) coincides with the measure algebra M(Rn) for 1 < p < 1 and 1 ? q < 1.

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35

Castejón, A., E. Corbacho, and V. Tarieladze. "AMD-Numbers, Compactness, Strict Singularity and the Essential Spectrum of Operators." gmj 9, no. 2 (June 2002): 227–70. http://dx.doi.org/10.1515/gmj.2002.227.

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Abstract For an operator 𝑇 acting from an infinite-dimensional Hilbert space 𝐻 to a normed space 𝑌 we define the upper AMD-number and the lower AMD-number as the upper and the lower limit of the net (δ(𝑇|𝐸))𝐸∈𝐹𝐷(𝐻), with respect to the family 𝐹𝐷(𝐻) of all finite-dimensional subspaces of 𝐻. When these numbers are equal, the operator is called AMD-regular. It is shown that if an operator 𝑇 is compact, then and, conversely, this property implies the compactness of 𝑇 provided 𝑌 is of cotype 2, but without this requirement may not imply this. Moreover, it is shown that an operator 𝑇 has the property if and only if it is superstrictly singular. As a consequence, it is established that any superstrictly singular operator from a Hilbert space to a cotype 2 Banach space is compact. For an operator 𝑇, acting between Hilbert spaces, it is shown that and are respectively the maximal and the minimal elements of the essential spectrum of , and that 𝑇 is AMD-regular if and only if the essential spectrum of |𝑇| consists of a single point.

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36

Luisa Di Piazza. "Kurzweil-Henstock Type Integration on Banach Spaces." Real Analysis Exchange 29, no. 2 (2004): 543. http://dx.doi.org/10.14321/realanalexch.29.2.0543.

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37

Knaust, Helmut. "Orlicz sequence spaces of Banach-Saks type." Archiv der Mathematik 59, no. 6 (December 1992): 562–65. http://dx.doi.org/10.1007/bf01194848.

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38

Salem, Hussein A. H. "Hadamard-type fractional calculus in Banach spaces." Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 113, no. 2 (March 30, 2018): 987–1006. http://dx.doi.org/10.1007/s13398-018-0531-y.

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39

Shukla, Satish. "Prešić type results in 2-Banach spaces." Afrika Matematika 25, no. 4 (June 21, 2013): 1043–51. http://dx.doi.org/10.1007/s13370-013-0174-2.

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40

Miana, Pedro J., Juan J. Royo, and Luis Sánchez-Lajusticia. "Convolution Algebraic Structures Defined by Hardy-Type Operators." Journal of Function Spaces and Applications 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/212465.

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The main aim of this paper is to show that certain Banach spaces, defined via integral kernel operators, are Banach modules (with respect to some known Banach algebras and convolution products onℝ+). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spacesLωpℝ+forp≥1. We also show new inequalities in these weighted Lebesgue spaces. These results are applied to several concrete function spaces, for example, weighted Sobolev spaces and fractional Sobolev spaces defined by Weyl fractional derivation.

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41

Caetano, António, Amiran Gogatishvili, and Bohumír Opic. "Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 5 (June 23, 2016): 905–27. http://dx.doi.org/10.1017/s0308210515000761.

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There are two main aims of the paper. The first is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second is to extend the criterion for the precompactness of sets in the Lebesgue spaces Lp(ℝn), 1 ⩽ p < ∞, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces , into Lorentz-type spaces.

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42

Hudzik, Henryk, Vatan Karakaya, Mohammad Mursaleen, and Necip Simsek. "Banach-Saks Type and Gurariǐ Modulus of Convexity of Some Banach Sequence Spaces." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/427382.

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43

Kim, Sang Og, and John Michael Michael Rassias. "Stability of the Apollonius Type Additive Functional Equation in Modular Spaces and Fuzzy Banach Spaces." Mathematics 7, no. 11 (November 17, 2019): 1125. http://dx.doi.org/10.3390/math7111125.

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In this work, we investigate the generalized Hyers-Ulam stability of the Apollonius type additive functional equation in modular spaces with or without Δ 2 -conditions. We study the same problem in fuzzy Banach spaces and β -homogeneous Banach spaces. We show the hyperstability of the functional equation associated with the Jordan triple product in fuzzy Banach algebras. The obtained results can be applied to differential and integral equations with kernels of non-power types.

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44

Rodríguez-López, Salvador, and Javier Soria. "A new class of restricted type spaces." Proceedings of the Edinburgh Mathematical Society 54, no. 3 (April 11, 2011): 749–59. http://dx.doi.org/10.1017/s0013091509001710.

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AbstractWe find new properties for the space R(X), introduced by Soria in the study of the best constant for the Hardy operator minus the identity. In particular, we characterize when R(X) coincides with the minimal Lorentz space Λ(X). The condition that R(X) ≠ {0} is also described in terms of the embedding (L1, ∞ ∩ L∞) ⊂ X. Finally, we also show the existence of a minimal rearrangement-invariant Banach function space (RIBFS) X among those for which R(X) ≠ {0} (which is the RIBFS envelope of the quasi-Banach space L1, ∞ ∩ L∞).

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45

Avilés, Antonio, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, and Yolanda Moreno. "On ultrapowers of Banach spaces of type \mathscrL∞." Fundamenta Mathematicae 222, no. 3 (2013): 195–212. http://dx.doi.org/10.4064/fm222-3-1.

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46

Park, Choon-Kil, Seong-Ki Hong, and Myoung-Jung Kim. "JENSEN TYPE QUADRATIC-QUADRATIC MAPPING IN BANACH SPACES." Bulletin of the Korean Mathematical Society 43, no. 4 (November 30, 2006): 703–9. http://dx.doi.org/10.4134/bkms.2006.43.4.703.

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47

Donchev, Tzanko. "Mixed type semicontinuous differential inclusions in Banach spaces." Annales Polonici Mathematici 77, no. 3 (2001): 245–59. http://dx.doi.org/10.4064/ap77-3-4.

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Kavadjiklis, Andreas, and Sung Guen Kim. "Plank type problems for polynomials on Banach spaces." Journal of Mathematical Analysis and Applications 396, no. 2 (December 2012): 528–35. http://dx.doi.org/10.1016/j.jmaa.2012.06.029.

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Ceng, L. C., A. Petruşel, and S. Y. Wu. "ON HYBRID PROXIMAL-TYPE ALGORITHMS IN BANACH SPACES." Taiwanese Journal of Mathematics 12, no. 8 (November 2008): 2009–29. http://dx.doi.org/10.11650/twjm/1500405133.

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Cascales, B. "A Krein-Smulian type result in Banach spaces." Quarterly Journal of Mathematics 48, no. 190 (June 1, 1997): 161–67. http://dx.doi.org/10.1093/qjmath/48.190.161.

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