Normal convergence implies both local normal convergence and compact normal convergence.
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Normal convergence implies both local normal convergence and compact normal convergence.
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Normal convergence implies both local normal convergence and compact normal convergence.
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And if the domain is locally compact ( even in the weakest sense ), then local normal convergence implies compact normal convergence.
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And if the domain is locally compact ( even in the weakest sense ), then local normal convergence implies compact normal convergence.
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For functions defined on a topological space, one can define ( as above ) local uniform convergence and compact normal convergence ( absolute convergence on compact sets ).
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As well, normal convergence of a series is different from " norm-topology convergence ", i . e . convergence of the partial sum sequence in the topology induced by the uniform norm.
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:To be honest I can't understand how this allegation could ever have been entertained as a serious possibility, since there is not the slightest shred of evidence to support it, apart from a perfectly normal convergence of views on certain topics.
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Normal convergence implies norm-topology convergence if and only iff the space of functions under consideration is complete with respect to the uniform norm . ( The converse does not hold even for complete function spaces : for example, consider the harmonic series as a sequence of constant functions ).
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