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Isolated points of some sets of bounded cosine families, bounded semigroups, and bounded groups on a Banach space

Isolated points of some sets of bounded cosine families, bounded semigroups, and bounded groups on a Banach space

We show that if the set of all bounded strongly continuous cosine families on a Banach space $X$ is treated as a metric space under the metric of the uniform convergence associated with the operator norm on the space $\mathcal {L}(X)$ of all bounded linea