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seratend is offline
Jan11-05, 07:34 AM
P: 318
The answer:

There is no compact (continuous) injective operator on a non separable hilbert space.

Quick demo: H the hilbert space. T the compact injective operator => Closure[T(H)] is a separable space (compacity property) => there is no injection between a non separable hilbert space and the separable one.

Moreover, we have a weaker formulation of this impossibility: there is no injective compact operator between a non separable Hilbert space and any banach space. (i.e. we can have an injective compact operator between 2 banach spaces, even if the banach spaces are non separable).