Note on the weak polynomial topology

S.V.Sharyn; A.V.Zagorodnyuk

We prove that there is a net $\{x_\alpha\}$ from the unit sphere of a complex infinite dimensional Banach space such that $\{\lambda_\alpha x_\alpha\}$ is convergent to zero in weak polynomial topology for an arbitrary collection of numbers $\{\lambda_\alpha\}$. Also we consider the situation in the real case.

1. R. Aron, B. Cole and T. Gamelin. Spectra of algebras of analytic functions on a Banach space// J. Reine Angew. Math. -- 1991. -- Bd.415. -- S.51--93.

2. R. Aron and P. Hájec. Zero sets of polynomials in several variables// Arhiv der Mathematic. -- 2006. -- V.56, № 6. -- P.561--568.

3. A. M Davie and T. W. Gamelin. A theorem on polynomial-star approximation// Proc. Amer. Math. Soc. -- 1989. -- V.106, № 2. -- P.351--356.

4. S. Dineen. Complex Analysis on Infinite Dimensional Spaces. -- London: Springer Verlag, 1999.

5. M. González, J. Gutiérrez and J. Llavona. Polynomial contiuity on $l_1$// Proc. Amer. Math. Soc. -- 1997. -- V.125, № 5. -- P.1349--1353.

6. A. Plichko and A. Zagorodnyuk. On automatic continuity and three problems of “The Scottish Book” concerning the boundedness of polynomial functionals// J. Math. Anal. Appl. -- 1998. -- V.220. -- P.477--494.

	Note on the weak polynomial topology
Author	S.V.Sharyn, A.V.Zagorodnyuk sharynsir@yahoo.com, andriyzag@yahoo.com Precarpatian National University, Ivano-Frankivsk, Ukraine, Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine,
Abstract	We prove that there is a net $\{x_\alpha\}$ from the unit sphere of a complex infinite dimensional Banach space such that $\{\lambda_\alpha x_\alpha\}$ is convergent to zero in weak polynomial topology for an arbitrary collection of numbers $\{\lambda_\alpha\}$. Also we consider the situation in the real case.
Keywords	weak polynomial topology, net, complex infinite dimensional Banach space
DOI	doi:10.30970/ms.30.1.98-100
Reference	1. R. Aron, B. Cole and T. Gamelin. Spectra of algebras of analytic functions on a Banach space// J. Reine Angew. Math. -- 1991. -- Bd.415. -- S.51--93. 2. R. Aron and P. Hájec. Zero sets of polynomials in several variables// Arhiv der Mathematic. -- 2006. -- V.56, № 6. -- P.561--568. 3. A. M Davie and T. W. Gamelin. A theorem on polynomial-star approximation// Proc. Amer. Math. Soc. -- 1989. -- V.106, № 2. -- P.351--356. 4. S. Dineen. Complex Analysis on Infinite Dimensional Spaces. -- London: Springer Verlag, 1999. 5. M. González, J. Gutiérrez and J. Llavona. Polynomial contiuity on $l_1$// Proc. Amer. Math. Soc. -- 1997. -- V.125, № 5. -- P.1349--1353. 6. A. Plichko and A. Zagorodnyuk. On automatic continuity and three problems of “The Scottish Book” concerning the boundedness of polynomial functionals// J. Math. Anal. Appl. -- 1998. -- V.220. -- P.477--494.
Pages	98-100
Volume	30
Issue	1
Year	2008
Journal	Matematychni Studii
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