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is bounded linear operator necessarily continuous? 3 This property is unrelated to the completeness of the domain or range, but instead only to the linear nature of the operator Yes, a linear operator (between normed spaces) is bounded if and only if it is continuous
Meaning of the continuous spectrum and the residual spectrum @Konstantin : The continuous spectrum requires that you have an inverse that is unbounded If X X is a complete space, then the inverse cannot be defined on the full space It is standard to require the inverse to be defined on a dense subspace If it is defined on a non-dense subspace, that falls into the miscellaneous category of residual
Proving the inverse of a continuous function is also continuous 6 All metric spaces are Hausdorff Given a continuous bijection between a compact space and a Hausdorff space the map is a homeomorphism Proof: We show that f f is a closed map Let K ⊂E1 K ⊂ E 1 be closed then it is compact so f(K) f (K) is compact and compact subsets of Hausdorff spaces are closed Hence, we have that f f is a homeomorphism