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 5h comment Prove that all three metrics induces the same topology on $X_1\times X_2$ You can prove that this induces in each case the product topology. 1d revised Composition involving bounded linear operators added 31 characters in body 1d comment Composition involving bounded linear operators Ok, I see. Your operator $U$ is not linear and indeed my argument only works for Banach spaces. 1d revised Composition involving bounded linear operators deleted 81 characters in body 1d answered Composition involving bounded linear operators Feb 2 awarded Notable Question Jan 28 comment A set of $n$ vectors is a basis if and only if… Probably the dimension of $V$ is fixed by $n$. Then a spanning set of $n$ vectors is linearly independent. Jan 23 answered How to show path-connectedness of $GL(n,\mathbb{C})$ Jan 15 answered If $\int_0^1 f x^n dx=0, \forall n \in \mathbb{N}$, then $f\equiv 0.$ Jan 13 comment connected sets and ordered sets Yes. The argument does not depend on the natural numbers. Jan 9 awarded Notable Question Dec 27 awarded Popular Question Dec 23 comment Prob 3 Sec 9 in Munkres TOPOLOGY 2nd ed: injections $f_n\colon\{1,\ldots,n\}\to A$ implies an injection $f \colon\mathbb{N}\to A$? I guess you need the axiom of (at least dependent) choice, to get $f$ since you define $f$ recursively. For the rest, the proof (and the idea) is right. Dec 16 comment Positive logarithm in a $C^*$-algebra Ok, thanks for confirmation. Dec 16 accepted Positive logarithm in a $C^*$-algebra Dec 14 comment Positive logarithm in a $C^*$-algebra That is true, so I guess the exercise is just wrong ? See here for reference: books.google.de/… Dec 14 revised Positive logarithm in a $C^*$-algebra added 179 characters in body Dec 14 revised Positive logarithm in a $C^*$-algebra edited tags Dec 14 comment Positive logarithm in a $C^*$-algebra The questions is about the existence of some positive logarithm ... This is exercise 3.6 in Conways Book on Operator Theory. Dec 14 asked Positive logarithm in a $C^*$-algebra