Operator Theory
Continuity refers to the property of a function where small changes in the input result in small changes in the output. In the context of linear operators, this concept plays a crucial role as it relates to boundedness, which ensures that the operator behaves predictably. When discussing compact self-adjoint operators, continuity is vital in analyzing the spectra and ensuring that limits of sequences of operators behave well. Additionally, continuity is essential in functional calculus as it allows for the extension of functions to operators. In dealing with unbounded linear operators, understanding continuity helps clarify how these operators act on their domains and how limits are approached within those domains.
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