Cardinalities of Restricted Ranges

Abstract

Let $l$ and $u$ be upper and lower semicontinuous extended functions on [a,b], respectively, with $l≤u$. Let $H$ be an n-dimensional Haar subspace and $K=\{p\in H:l\leq p\leq u\}$. This paper gives complete characterizations of K satisfying $$card \ K=0 \ or \ 1 \  or \ ∞$$ under certain assumptions, where card K denotes the cardinality of K.