All Questions
Tagged with axiom-of-choice fa.functional-analysis
22
questions
1
vote
0
answers
85
views
Everywhere-defined unbounded operators between Banach spaces
In this post, it is said that there are no constructive examples of everywhere-defined unbounded operators between Banach spaces; every example furnished must use the axiom of choice. This seems like ...
- 141
1
vote
0
answers
732
views
Finding a unique and finite expected value for almost all measurable functions?
Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...
- 1
14
votes
1
answer
1k
views
Stone-Weierstrass Theorem without AC
To what extent does the usual Stone-Weierstrass Theorem depend on some form of the Axiom of Choice? There seems to be a lot of literature on constructive versions in toposes, but I have been unable ...
- 2,019
4
votes
0
answers
242
views
A question regarding the Hahn-Banach theorem and Banach limits
Set theorists typically prove the existence of Banach limits (EBL) using the Ultrafilter Theorem or, its equivalent, the Boolean Prime Ideal Theorem (BPI). Analysts, on the other hand, typically prove ...
- 6,190
5
votes
0
answers
272
views
Completeness of the space $L^p$ and the Axiom of Countable Choice
I am thinking about the proof that the usual $L^p$ spaces are complete.
So, let $(X,\mathcal{F},\mu)$ be a measure space and let
$p\in[1,+\infty)$.
Important: by a measure I mean a nonnegative $\sigma$...
- 925
8
votes
2
answers
888
views
Continuous linear functionals and the Axiom of Choice
Can one prove without the Axiom of Choice that for every normed vector space $X$ there exist a nonzero continuous linear functional on $X$?
- 925
0
votes
1
answer
319
views
Can we choose a sequence of Hilbert spaces?
Let $n$ be a fixed natural number.
Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed subspaces of $H$.
Set $H_0\mathrel{:=}H_1\cap H_2\cap\dotsb\cap H_n$ and let
$P_i$ be the ...
- 925
0
votes
1
answer
463
views
Is a function needed here?
This question is related to my question Can we choose an element from a class?.
However, I decided to create a separate question.
Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed ...
- 925
2
votes
1
answer
450
views
Can we choose an element from a class?
Let $H$ be a complex Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$.
Set $H_0:=H_1\cap H_2\cap...\cap H_n$ and let
$P_i$ be the orthogonal projection onto $H_i$, $i=0,1,2,...,n$.
I study ...
- 925
12
votes
0
answers
460
views
Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?
Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...
- 80.3k
0
votes
0
answers
57
views
Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?
Suppose I have an equation in some reflexive separable Banach space $X$:
$$Au=f$$
for given data $f \in X^*$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...
- 107
20
votes
2
answers
1k
views
Can There be a 1 dimensional Banach-Tarski paradox in the absence of choice
Let $\mathbb{R}$ act on itself by translation. Then there is no finite decomposition of a unit interval into pieces which, when translated, yields two distinct unit intervals.
More formally does ...
- 545
11
votes
1
answer
736
views
Generalized limits on $\ell^\infty(\mathbb{N})$
Let $\ell^\infty(\mathbb{N})$ denote the set of bounded real sequences $(a_n)_{n\in\mathbb{N}}$. The $\lim$ operator is a partial linear operator from $\ell^\infty(\mathbb{N})$ to $\mathbb{R}$. With ...
4
votes
1
answer
583
views
Construction of a codimension 1 dense subspace without Zorn
Suppose $X$ is an infinite dimensional topological vector space
and $v\in X$ is non-zero. It is then not difficult to construct
a vector space $U\subset X$ so that
1) $U$ is dense in $X$.
2) $U+{...
- 462
11
votes
1
answer
298
views
Without AC, which implications between the different definitions of amenability still hold?
More precisely, I would like to know which implications between the following definitions of amenability of a discrete countable (or even finitely generated) group can be proved to hold with only ZF (...
- 402