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1 results

  • 12 - Completeness of the Space of Operators

  • Adam Bobrowski, Politechnika Lubelska, Poland
  • Book:
    Functional Analysis Revisited
    Published online:
    31 October 2024
    Print publication:
    04 July 2024, pp 127-143

  • Summary

    Bounded linear operators form a natural class of maps between normed linear spaces: they are, by definition, linear and continuous, so that, in other words, they preserve, at least to some extent, the linear and the topological structures of the normed spaces involved. As it turns out, bounded linear operators from one normed linear space to another form a normed linear space themselves. Moreover, if the range space is complete, the space of operators is complete also, and is thus a Banach space. We note that we have already encountered examples of bounded linear operators on the previous pages of this book and discuss a score of new ones. Nor do we refrain from calculating norms of some of them.