On extension of absolutely continuous functions and isometries on the normed spaces

چکیده مقاله

1. Suppose that  $X$ and $Y$ are  normed spaces and let $ F $ and $ G $ be the functions on $ X$ such that are of bounded variation and bounded away from zero. We show that the summation of their inverses is also, on the desired space, of bounded variation.  Also, suppose that $X\subseteq\mathbb{R}$ is bounded and $F:X\longrightarrow\mathbb{R}$ is an absolutely continuous map, we prove that $F$ has a unique uniformly continuous extension of bounded variation. Given a completely regular space and its Stone-Čech compactification, we prove that every bounded continuous real function on the mentioned space can be uniquely extended to a continuous real function on the Stone-Čech compactification. Considering a surjective linear isometry between the absolutely continuous and bounded spaces $AC_b(X) $ and $ AC_b(Y) $, we show that there exists a monotone homomorphism between the closures of two spaces $ Y $ and $ X $, say $ \overline{Y} $ and $ \overline{X} $.