# Algebras of real analytic functions: Homomorphisms and bounding sets

Peter Biström; Jesús Jaramillo; Mikael Lindström

Studia Mathematica (1995)

- Volume: 115, Issue: 1, page 23-37
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topBiström, Peter, Jaramillo, Jesús, and Lindström, Mikael. "Algebras of real analytic functions: Homomorphisms and bounding sets." Studia Mathematica 115.1 (1995): 23-37. <http://eudml.org/doc/216196>.

@article{Biström1995,

abstract = {This article deals with bounding sets in real Banach spaces E with respect to the functions in A(E), the algebra of real analytic functions on E, as well as to various subalgebras of A(E). These bounding sets are shown to be relatively weakly compact and the question whether they are always relatively compact in the norm topology is reduced to the study of the action on the set of unit vectors in $l_∞$ of the corresponding functions in $A(l_∞)$. These results are achieved by studying the homomorphisms on the function algebras in question, an idea that is also reversed in order to obtain new results for the set of homomorphisms on these algebras.},

author = {Biström, Peter, Jaramillo, Jesús, Lindström, Mikael},

journal = {Studia Mathematica},

keywords = {bounding sets in real Banach spaces; relatively weakly compact; homomorphisms; function algebras},

language = {eng},

number = {1},

pages = {23-37},

title = {Algebras of real analytic functions: Homomorphisms and bounding sets},

url = {http://eudml.org/doc/216196},

volume = {115},

year = {1995},

}

TY - JOUR

AU - Biström, Peter

AU - Jaramillo, Jesús

AU - Lindström, Mikael

TI - Algebras of real analytic functions: Homomorphisms and bounding sets

JO - Studia Mathematica

PY - 1995

VL - 115

IS - 1

SP - 23

EP - 37

AB - This article deals with bounding sets in real Banach spaces E with respect to the functions in A(E), the algebra of real analytic functions on E, as well as to various subalgebras of A(E). These bounding sets are shown to be relatively weakly compact and the question whether they are always relatively compact in the norm topology is reduced to the study of the action on the set of unit vectors in $l_∞$ of the corresponding functions in $A(l_∞)$. These results are achieved by studying the homomorphisms on the function algebras in question, an idea that is also reversed in order to obtain new results for the set of homomorphisms on these algebras.

LA - eng

KW - bounding sets in real Banach spaces; relatively weakly compact; homomorphisms; function algebras

UR - http://eudml.org/doc/216196

ER -

## References

top- [1] R. Alencar, R. Aron and S. Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90 (1984), 407-411. Zbl0536.46015
- [2] P. Biström, S. Bjon and M. Lindström, Function algebras on which homomorphisms are point evaluations on sequences, Manuscripta Math. 73 (1991), 179-185. Zbl0764.46048
- [3] P. Biström and J. A. Jaramillo, ${C}^{\infty}$-bounding sets and compactness, Math. Scand. 75 (1994), 82-86.
- [4] J. Bochnak and J. Siciak, Analytic functions in topological vector spaces, Studia Math. 39 (1971), 77-112. Zbl0214.37703
- [5] J. Bourgain and J. Diestel, Limited operators and strict cosingularity, Math. Nachr. 119 (1984), 55-58. Zbl0601.47019
- [6] T. K. Carne, B. Cole and T. W. Gamelin, A uniform algebra of analytic functions on a Banach space, Trans. Amer. Math. Soc. 314 (1989), 639-659. Zbl0704.46033
- [7] J. Castillo and C. Sánchez, Weakly-p-compact, p-Banach-Saks and super-reflexive Banach spaces, J. Math. Anal. Appl., to appear. Zbl0878.46009
- [8] K. Ciesielski and R. Pol, A weakly Lindelöf function space C(K) without any continuous injection into ${c}_{0}\left(\Gamma \right)$, Bull. Polish Acad. Sci. Math. 32 (1984), 681-688. Zbl0571.54014
- [9] A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), 351-356. Zbl0683.46037
- [10] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math. 92, Springer, 1984.
- [11] K. Floret, Weakly Compact Sets, Lecture Notes in Math. 801, Springer, 1980. Zbl0437.46006
- [12] P. Galindo, D. García, M. Maestre and J. Mujica, Extension of multilinear mappings on Banach spaces, Studia Math. 108 (1994), 55-76. Zbl0852.46004
- [13] M. Garrido, J. Gómez and J. A. Jaramillo, Homomorphisms on function algebras, Extracta Math. 7 (1992), 46-52. Zbl0808.46035
- [14] T. Husain, Multiplicative Functionals on Topological Algebras, Research Notes in Math. 85, Pitman, 1983. Zbl0514.46030
- [15] J. A. Jaramillo and A. Prieto, The weak-polynomial convergence on a Banach space, Proc. Amer. Math. Soc. 118 (1993), 463-468. Zbl0795.46042
- [16] H. Jarchow, Locally Convex Spaces, Teubner, 1981. Zbl0466.46001
- [17] K. John, H. Toruńczyk and V. Zizler, Uniformly smooth partitions of unity on superreflexive Banach spaces, Studia Math. 70 (1981), 129-137. Zbl0485.46006
- [18] B. Josefson, Bounding subsets of ${l}_{\infty}\left(A\right)$, J. Math. Pures Appl. 57 (1978), 397-421. Zbl0403.46011
- [19] A. Kriegl and P. Michor, More smoothly real compact spaces, Proc. Amer. Math. Soc. 117 (1993), 467-471. Zbl0813.46026
- [20] J. G. Llavona, Approximation of Continuously Differentiable Functions, North-Holland, 1986. Zbl0642.41001
- [21] J. Mujica, Complex homomorphisms of the algebras of holomorphic functions on Fréchet spaces, Math. Ann. 241 (1979), 73-82. Zbl0386.46023
- [22] S. Negrepontis, Banach spaces and topology, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, 1984, 1047-1142.
- [23] T. Schlumprecht, A limited set that is not bounding, Proc. Roy. Irish Acad. 90A (1990), 125-129. Zbl0749.46030

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.