Convergence and Stability

Carlos S. Kubrusly
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Carlos S. Kubrusly: Catholic University of Rio de Janeiro

Chapter 3 in Hilbert Space Operators, 2003, pp 23-32 from Springer

Abstract: Abstract Let χ and у be formed spaces and let {T n } be a В[χ, у]-valued sequence (i.e., a sequence of transformations in В [χ, у]). If {T n } converges in the formed space B[χ, у];that is, if there exists T in B [χ, у] such that $$\left\| {{T_n} - T} \right\| \to 0,$$ then we say that {T n } converges uniformly to T. This (unique) T ∈ B[χ, у] is called the uniform limit of {T n }. Notation: $${T_n}\xrightarrow{u}T$$ . If {T n } does not converge uniformly to T, then we write $${T_n}\mathop {\not \to }\limits^u T$$ . The у -valued sequence {T n х} converges in у for every x in χ if and only if there exists a (unique) linear transformation T of χ into у such that $$\left\| {({T_n} - T)x} \right\| \to 0 for every x \in \chi .$$

Date: 2003
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DOI: 10.1007/978-1-4612-2064-0_3

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