## Rocky Mountain Journal of Mathematics

- Rocky Mountain J. Math.
- Volume 49, Number 6 (2019), 1829-1842.

### Bases in the spaces of homogeneous polynomials and multilinear operators on Banach spaces

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#### Abstract

For Banach spaces $E_1, \dots ,E_m$, $E$ and $F$ with their bases, we show that a particular monomial sequence forms a basis of $\mathcal {P}(^mE; F)$, the space of continuous $m$-homogeneous polynomials from $E$ to $F$ (resp.\ a basis of $\mathcal {L}(E_1,\dots ,E_m;F)$, the space of continuous $m$-linear operators from $E_1\times \cdots \times E_m$ to $F$) if and only if the basis of $E$ (resp. the basis of $E_1,\dots ,E_m$) is a shrinking basis and every $P \in \mathcal {P}(^mE; F)$ (resp.\ every $T \in \mathcal {L}(E_1,\dots ,E_m;F)$) is weakly continuous on bounded sets.

#### Article information

**Source**

Rocky Mountain J. Math., Volume 49, Number 6 (2019), 1829-1842.

**Dates**

First available in Project Euclid: 3 November 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.rmjm/1572746420

**Digital Object Identifier**

doi:10.1216/RMJ-2019-49-6-1829

**Mathematical Reviews number (MathSciNet)**

MR4027235

**Subjects**

Primary: 46G25: (Spaces of) multilinear mappings, polynomials [See also 46E50, 46G20, 47H60]

Secondary: 46M05: Tensor products [See also 46A32, 46B28, 47A80] 46B28: Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]

**Keywords**

homogeneous polynomials multilinear operators monomial bases symmetric tensor products

#### Citation

Ji, Donghai; Bu, Qingying. Bases in the spaces of homogeneous polynomials and multilinear operators on Banach spaces. Rocky Mountain J. Math. 49 (2019), no. 6, 1829--1842. doi:10.1216/RMJ-2019-49-6-1829. https://projecteuclid.org/euclid.rmjm/1572746420

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