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Results tagged with fa.functional-analysis
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user 1162
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
11
votes
1
answer
624
views
Inequivalent complete norms and the axiom of choice
Hi,
I've been wondering about the following :
Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space?
All the examples of inequivalent complete norms …
- 2,154
2
votes
$\mathcal P(K)=\mathcal R(K)$ iff $\Bbb C\backslash K$ is connected
Perhaps a more elementary proof is the following :
Assume $\mathbb{C} \setminus K$ is disconnected. Let $V$ be a bounded component of $\mathbb{C} \setminus K$, and let $w \in V$. Let us show that the …
- 2,154
5
votes
In a Banach algebra, do ab and ba have almost the same exponential spectrum?
Bonjour Yemon,
Oui, c'est un problème proposé par TJR dans le cadre d'un projet de recherche d'été CRSNG.
It is indeed a very subtle question, I thought it might interest some people here. This probl …
- 2,154
7
votes
Reference for complex analysis jargon
I really recommend the book Potential Theory in the complex plane"
by Thomas Ransford.
It's a very nice book with exercises and it covers each of the 5 points you mentioned.
- 2,154
21
votes
2
answers
2k
views
In a Banach algebra, do ab and ba have almost the same exponential spectrum?
Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is th …
- 2,154