A group invariant Bishop-Phelps theorem

摘要

We show that for any Banach space and any compact topological group $ Gsubset L(X)$ such that the norm of $ X$ is $ G$-invariant, the set of norm attaining $ G$-invariant functionals on $ X$ is dense in the set of all $ G$-invariant functionals on $ X$, where a mapping $ f$ is called $ G$-invariant if for every $ xin X$ and every $ gin G$, $ fig (g(x)ig )=f(x)$. In contrast, we show also that the analog of Bollobás result does not hold in general. A version of Bollobás and James' theorems is also presented.