Banach空间的双线性连续映射及其张量积

BILINEAR CONTINUOUS MAPS AND THEIR TENSOR PRODUCTS OF BANACH SPACES

设E,E′,E″与F,F′,F″是Banach空间(以下简称B空间),φ:E×E′→E″与ψ:F×F′→F″是双线性连续映射。本文通过一系列命题证明了由φ与ψ可以唯一地决定一个双线性连续映射ω:(E_vF)×(E′_vF′)→E″_vF″,其中v是Cross范数,F′_vF′,E_vF与E″_vF″分别是在v下E′_vF′,E_vF与E″_vF″的完备化。

Suppose E, E′, E″ and F, F′, F″ are Banach spaces (briefly called B-spaces). Letφ: E×E′→E″ andψ: F×F×F′→F″ be bilinear continuous maps, an important conclusion has been derived from several propositions that ifν is a cross-norm, φ and ψ can uniquely determine a bilinear continuous mapω: (EνF)×(E′νF′)→E″νF″, where EνF, E′νF′ and E″νF″ are completions, of EF′, E′F′ and E″F″ under the cross-norm ν, respectively.