Category:Definitions/Uniformly Continuous Semigroups

This category contains definitions related to Uniformly Continuous Semigroups.
Related results can be found in Category:Uniformly Continuous Semigroups.

Let $\GF \in \set {\R, \C}$.

Let $X$ be a Banach space over $\GF$.

Let $\family {\map T t}_{t \ge 0}$ be a $\hointr 0 \infty$-indexed family of bounded linear transformations $\map T t : X \to X$.

Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the space of bounded linear transformations equipped with the canonical norm.

We say that $\family {\map T t}_{t \ge 0}$ is uniformly continuous if and only if:

$\ds \lim_{t \mathop \to 0^+} \norm {\map T t - I}_{\map B X} = 0$