All Questions
Tagged with fa.functional-analysis fa.functional-analysis or
9,772 questions
32
votes
11
answers
23k
views
A book for problems in Functional Analysis
I want to know if there's any book that categorizes problems by subjects of Functional Analysis.
I'm studying Functional Analysis now a days and I really need to solve some problems in order to ...
32
votes
6
answers
3k
views
Can distribution theory be developed Riemann-free?
I imagine most people who frequent MO have been indoctrinated into the point of view that the Riemann integral can be safely discarded once one has taken the time to develop the Lebesgue integral. ...
- 29.2k
32
votes
3
answers
3k
views
Why are there so many fractional derivatives?
I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.
I started with the book The Fractional Calculus ...
- 3,738
32
votes
2
answers
4k
views
Are there non-reflexive vector spaces isomorphic to their bi-dual?
Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.
Are ...
- 7,902
31
votes
7
answers
4k
views
Intuition for failure of Implicit Function theorem on Frechet Manifolds
When dealing with moduli spaces of, say connections or metrics, I am using the notions of Frechet spaces/manifolds/groups. I have become familiar with Banach manifolds (I think), but Frechet manifolds ...
- 17.5k
31
votes
2
answers
3k
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Is a normed space which is homeomorphic to a Banach space complete?
I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.
Does this imply that $(E,||\cdot||)$ is also a Banach space?
I think I read something ...
- 495
31
votes
3
answers
5k
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When is an integral transform trace class?
Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator
$$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$
the operator $K$ is Hilbert ...
- 11.2k
31
votes
2
answers
1k
views
Open problems in Sobolev spaces
What are the open problems in the theory of Sobolev spaces?
I would like to see problems that are yes or no only. Also I would like to see problems with the statements that are short and easy to ...
31
votes
1
answer
2k
views
Szőkefalvi-Nagy's unitarizability theorem in the Calkin algebra?
Here's a research problem, which I think interesting.
Suppose that $t$ is an invertible element in the Calkin algebra $\mathcal{Q} = \mathcal{B}(\ell_2)/\mathcal{K}(\ell_2)$ which satisfies $\sup_{n \...
- 10.1k
31
votes
1
answer
2k
views
Topology on space of hyperfunctions
This is a reference request, coming from someone with little knowledge of hyperfunctions:
Which methods have been used to endow the space of hyperfunctions $\mathcal B(\mathbb R)$ with something like ...
- 21.3k
31
votes
0
answers
2k
views
Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?
Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...
- 4,878
31
votes
0
answers
1k
views
When are two C*-algebras isomorphic as Banach spaces?
We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its ...
- 3,497
30
votes
4
answers
4k
views
Elementary applications of Krein-Milman
This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try.
Recall that the Krein-...
30
votes
3
answers
3k
views
Surjectivity of operators on $\ell^\infty$
Can anyone give me an example of an bounded and linear operator $T:\ell^\infty\to \ell^\infty$ (the space of bounded sequences with the usual sup-norm), such that T has dense range, but is not ...
- 301
30
votes
1
answer
1k
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Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments
It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
- 6,853
29
votes
15
answers
6k
views
Important results that use infinite-dimensional manifolds?
Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn up? Important ...
29
votes
6
answers
9k
views
Nonseparable Hilbert spaces
Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces ...
- 9,330
29
votes
1
answer
4k
views
Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...
- 25.5k
28
votes
6
answers
6k
views
Any real contribution of functional analysis to quantum theory as a branch of physics?
In the last paragraph of this last paper of Klaas Landsman, you can read:
Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...
28
votes
6
answers
12k
views
Almost orthogonal vectors
This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
- 18.7k
28
votes
9
answers
5k
views
Applications of algebra to analysis
EDIT: I would like to make a list of modern applications of algebra in analysis. By "modern" I will mean developments since the beginning of the 20th century. It is well known that classical linear ...
28
votes
3
answers
4k
views
A separable Banach space and a non-separable Banach space having the same dual space?
I asked myself the following question when I was student just for curiosity. I asked a bit around (my professor, some researchers that I know), but nobody was able to give me an answer. So maybe it is ...
- 6,034
28
votes
7
answers
13k
views
Regular borel measures on metric spaces
When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
- 18.7k
28
votes
2
answers
3k
views
Intuition about L^p spaces
I have read somewhere the following very nice intuition about $L^p(\mathbb{R})$ spaces.
This graphic shows a lot of nice relations:
1) There is no inclusion between $L^p$ and $L^q$
2) $L^p$ is the ...
- 305
28
votes
2
answers
1k
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Can an operator have Exp(z) as its characteristic "polynomial"?
Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator. Define
$$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$
the ...
- 5,898
28
votes
2
answers
1k
views
What is the Banach-Mazur distance between $\ell_\infty$ and $L_\infty$?
Given Banach spaces $X$ and $Y$, the Banach-Mazur distance between $X$ and $Y$ is defined as
$$ d(X,Y) = \inf\{ \|\varphi\|\|\varphi^{-1}\| : \varphi\colon X\to Y \text{ isomorphism} \}.
$$
We ...
- 3,497
28
votes
2
answers
2k
views
Dynamical properties of injective continuous functions on $\mathbb{R}^d$
Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all $x\in\...
- 419
28
votes
1
answer
956
views
Grothendieck's in-spirit-category-theoretic functional analysis?
I heard several times (for instance in these general lectures) that Grothendieck did functional analysis before he started doing algebraic geometry and category theory. It is said that at the time he ...
- 333
27
votes
5
answers
3k
views
Nice applications for Schwartz distributions
I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are:
Some multilinear algebra including the Kernel Theorem and ...
27
votes
2
answers
8k
views
Compact embeddings of Sobolev spaces: a counterexample showing the Rellich-Kondrachov theorem is sharp
Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...
- 309
27
votes
4
answers
8k
views
Proofs of Young's inequality for convolution
For $1\leq p,q \leq \infty$ such that $\frac1p +\frac1q\geq 1$, Young's inequality states $\|f\star g\|_r\leq \|f\|_p\|g\|_q$ (we work on $\mathbf{R}^d$ here), where $1+\frac1r = \frac1p+\frac1q$. ...
- 3,425
27
votes
3
answers
5k
views
Weak and Strong Integration of vector-valued functions
This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...
- 741
27
votes
1
answer
1k
views
Do Sobolev spaces contain nowhere differentiable functions?
Does the Sobolev space $H^1(R^n)$ of weakly differentiable functions on a bounded domain in $R^n$ (or a more general Sobolev space) contain a continuous but nowhere differentiable function?
- 3,873
27
votes
2
answers
5k
views
What can be said about the Fourier transforms of characteristic functions?
What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular,
What properties are common to ...
- 2,719
27
votes
1
answer
4k
views
Criteria for boundedness of power series
Consider a power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all real
x, thus defining a function $f: \mathbb{R} \to \mathbb{R}$.
Can one give necessary and sufficient criteria the ...
- 4,014
27
votes
1
answer
1k
views
The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
- 683
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
26
votes
6
answers
8k
views
prime ideals in C([0,1])
It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.
So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, ...
- 5,063
26
votes
3
answers
16k
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the dual space of C(X) (X is noncompact metric space)
It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f:
X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of ...
- 1,706
26
votes
2
answers
5k
views
Does Arzelà-Ascoli require choice?
Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...
- 29.8k
26
votes
4
answers
5k
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Can $L^{2}$ be represented as a space of functions (not equivalence classes)?
Let $X$ be the vector space of all Lebesgue-measurable functions $f:\left[a,b\right]\rightarrowℝ$ such that $\int^{b}_{a}\left|f\left(x\right)\right|^{2}dx<\infty$ (Lebesgue integral). Then we can ...
- 4,597
26
votes
2
answers
6k
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Understanding a simplifying assumption in proof of the invariant subspace problem
In a recent preprint On the invariant subspace problem in Hilbert spaces Per H. Enflo claims to have solved the invariant subspace problem, showing that every bounded linear operator on a separable ...
- 423
26
votes
3
answers
2k
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About the category of von neumann algebras
I am looking for one (or more) reference about properties of the category of von Neumann algebra.
More precisely, in an answer of a previous question, Dmitri Pavlov mentions
that the $W^*$ category ...
- 357
26
votes
3
answers
7k
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Dual of bounded uniformly continuous functions
Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?
I ...
- 29.8k
26
votes
2
answers
3k
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Corollaries of the Yoneda Lemma in Analysis?
This is a cross-post of my ~2 weeks (canonically) unanswered question on Math.SE: https://math.stackexchange.com/questions/1830287/corollaries-of-the-yoneda-lemma-in-analysis.
I am looking for some ...
- 2,680
26
votes
2
answers
1k
views
Origin and first uses of $\ell_p$ norms?
When exactly were $\ell_p$ norms first defined and used?
(Here is what I know, or think I know: Lebesgue and/or Riesz had something to do with them, but in some sense they go back to Minkowski, since ...
- 20.2k
26
votes
3
answers
2k
views
Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
- 7,127
26
votes
1
answer
820
views
The maximal "nearly convex" function
The following problem is only tangentially related to my present work, and I do
not have any applications. However, I am curious to know the solution -- or
even to see a lack thereof, indicating that ...
- 23k
26
votes
2
answers
1k
views
Symmetric strengthening of the Cauchy-Schwarz inequality
In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have
\begin{align*}
\|v^2\| \, \|w^2\| - \langle ...
- 12.5k
26
votes
2
answers
2k
views
When is a locally convex topological vector space normal or paracompact?
All locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is ...
- 26.8k