zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On holomorphic functions attaining their norms. (English) Zbl 1086.46034
Authors’ abstract: We show that on a complex Banach space $X$, the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball attaining their norms are dense provided that $X$ has the Radon-Nikodym property. We also show that the same result holds for Banach spaces having the strengthened version of the approximation property but considering just functions with are also weakly uniformly continuous on the unit ball. We prove that there exists a polynomial such that for any fixed positive integer $k,$ it cannot be approximated by norm attaining polynomials with degree less than $k.$ For $X = d_{w^*}(w,1),$ a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal to two attains its norm if, and only if, each polynomial attains its norm.

MSC:
46G25(Spaces of) multilinear mappings, polynomials
46B25Classical Banach spaces in the general theory of normed spaces
46B03Isomorphic theory (including renorming) of Banach spaces
46G20Infinite dimensional holomorphy
WorldCat.org
Full Text: DOI
References:
[1] Acosta, M. D.: On multilinear mappings attaining their norms. Studia math. 131, 155-165 (1998) · Zbl 0934.46048
[2] Acosta, M. D.; Aguirre, F.; Payá, R.: There is no bilinear Bishop--phelps theorem. Israel J. Math. 93, 221-227 (1996) · Zbl 0852.46010
[3] M.D. Acosta, J. Alaminos, D. Garcı\acute{}a, M. Maestre, Perturbed optimization principle for dual spaces with the Radon--Nikodym property, preprint
[4] Aron, R.; Cole, B.; Gamelin, T. W.: Weak-star continuous analytic functions. Canad. J. Math. 47, 673-683 (1995) · Zbl 0829.46034
[5] Aron, R. M.; Boyd, C.; Choi, Y. S.: Unique Hahn--Banach theorems for spaces of homogeneous polynomials. J. austral. Math. soc. 70, 387-400 (2001) · Zbl 1036.46032
[6] Aron, R. M.; Prolla, J. B.: Polynomial approximation of differentiable functions on Banach spaces. J. reine angew. Math. 313, 195-216 (1980) · Zbl 0413.41022
[7] Y.S. Choi, D. Garcı\acute{}a, S.G. Kim, M. Maestre, Norm or numerical radius attaining polynomials on C(K), J. Math. Anal. Appl., in press · Zbl 1059.46026
[8] Y.S. Choi, K.H. Han, H.G. Song, Extensions of polynomials on preduals of Lorentz sequence spaces, preprint · Zbl 1092.46030
[9] Choi, Y. S.; Kim, S. G.: Norm or numerical radius attaining multilinear mappings and polynomials. J. London math. Soc. 54, 135-147 (1996) · Zbl 0858.47005
[10] Dineen, S.: Complex analysis on infinite dimensional spaces. Springer monographs in math. (1999)
[11] Finet, C.; Georgiev, P.: Optimization by n-homogeneous polynomial perturbations. Bull. soc. Roy. sci. Liège 70, 251-257 (2001) · Zbl 1005.90063
[12] Jiménez-Sevilla, M.; Payá, R.: Norm attaining multilinear forms and polynomials on predual of Lorentz sequence spaces. Studia math. 127, 99-112 (1998) · Zbl 0909.46015
[13] Johnson, J.; Wolfe, J.: Norm attaining operators. Studia math. 65, 7-19 (1979) · Zbl 0432.47024
[14] A. Kamińska, H.J. Lee, On uniqueness of extension of homogeneous polynomials, preprint · Zbl 1106.46030
[15] Lindenstrauss, J.; Tzafriri, L.: Classical Banach spaces I. (1977) · Zbl 0362.46013
[16] Stegall, C.: Optimization and differentiation in Banach spaces. Linear algebra appl. 84, 191-211 (1986) · Zbl 0633.46042
[17] Pełczyński, A.: A property of multilinear operations. Studia math. 16, 173-182 (1957) · Zbl 0080.09701
[18] Werner, D.: New classes of Banach spaces which are M-ideals in their biduals. Math. proc. Cambridge philos. Soc. 111, 337-354 (1992) · Zbl 0787.46020