On certain uniformly open multilinear mappings

Abstract

We obtain two results stating the uniform openness of bilinear operators and multilinear functionals. The first result deals with Banach spaces $L^p:=L_{\mathbb{K}}^p$ (over $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$) and pointwise multiplication from $L^p\times L^q$ to $L^r$ (where $1/p+1/q=1/r$). The second result is concerned with the nontrivial $n$-linear functionals from the product $X_1\times \cdots \times X_n$ of normed spaces (over $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$) to the field $\mathbb{K}$.