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Results tagged with fa.functional-analysis
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user 7206
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
18
votes
Accepted
Non continuous Linear form on $E=C([0,1],\mathbb{R})$ without AC
No. It's impossible.
In certain models of $\sf ZF+DC$ there is a property known as "automatic continuity" for Banach spaces, that means that every linear operator to a normed space is continuous.
Su …
- 39.8k
14
votes
Continuous linear functionals and the Axiom of Choice
No, this is equivalent to the Hahn–Banach theorem already in the case of Banach spaces.
See in my write up: https://arxiv.org/abs/2010.15632
- 39.8k
4
votes
Can we choose an element from a class?
It is often tempting to think that all our existential instantiations happen "in advance", a sort of proof theoretic "mise en place" if you will.
But they don't, and they don't have to be. For a given …
- 39.8k
2
votes
Cardinality of $C^*([0,1])$
Recall that $[0,1]$ is compact and therefore $C([0,1])$ is a Polish space. By standard arguments, there are only $2^{\aleph_0}$ continuous functions between two Polish spaces.
Therefore there are at …
- 39.8k