All Questions
Tagged with fa.functional-analysis fa.functional-analysis or
9,772 questions
9
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5
answers
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Convexity of distance-to-boundary function
Let $\Omega\subset\mathbb{R}^{n}$ be an open,
bounded convex domain. Denote $d_{\Omega}:\Omega\rightarrow\mathbb{R}$
the distance-to-boundary function, that is,
$$
d_{\Omega}\left(x\right):=\inf\left\...
- 91
9
votes
2
answers
758
views
Number of critical points of smooth functions on $S^1$
Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?
9
votes
4
answers
4k
views
Is the space of Radon measures a Polish space or at least separable?
Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
- 657
9
votes
1
answer
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Is a manifold generically real analytic (with generic real analytic metric)?
I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
- 123
9
votes
3
answers
568
views
Defining the abstract tensor product of W*-algebras via a universal property
I am playing around a bit with $W^*$-algebras, and I'm trying to come up with a definition for the $W^*$-algebraic tensor product. Here is my first attempt:
It is easy to show that such an object ...
- 175
9
votes
4
answers
2k
views
Books about capacity theory
While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...
- 2,222
9
votes
2
answers
334
views
Does $\pm A \leq B$ imply that $B^{-1} A$ is bounded?
Lately I have to use a lot of functional calculus. A question that keeps popping up and that I don't manage to resolve is the following:
Let $A,B$ be self-adjoint (not necessarily bounded) operators ...
- 1,045
9
votes
4
answers
905
views
Defining the value of a distribution at a point
Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
- 1,942
9
votes
3
answers
735
views
Cyclic vectors for the shift operator
Let $S:\ell^2\to\ell^2$ be the shift operator
$$
S(x_1,x_2,\dots)=(0,x_1,x_2,\dots).
$$
Let $x\in \ell^2$ with $x_1=1$. Is $x$ cyclic for $S$? In other words, is the span of the vectors $x,Sx,S^2x,\...
user1688
9
votes
2
answers
298
views
Two inequalities in $C^*$ algebras
Under what conditions on a $C^*$ algebra $A$ we have the following inequality:
$$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$
The second identity which I am looking for is ...
- 356
9
votes
3
answers
3k
views
The classical Krein-Rutman theorem
The classical Krein-Rutman theorem states that any positive compact linear endomorphism $T:X \to X$ on a Banach space $X$ with positive spectral radius $r(T)$ has an eigenvalue $r(T)$ with a positive ...
- 1,202
9
votes
2
answers
611
views
When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ hold?
Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert_{\mathrm{F}}=\sqrt{\sum_{i,j} A_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert_{\mathrm{op}}=\sup_{x \not = 0} \frac{\...
- 93
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3
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Projections in Banach spaces
Dear All,
I am absolutely lost in the following problem:
Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm min}...
- 93
9
votes
2
answers
394
views
Can one detect a cyclic and separating vector for a concrete $C^*$-algebra using a dense subalgebra?
Let $A$ be a $C^*$ algebra of operators acting on some Hilbert space $H$, and $A_0$ is a norm dense $*$-subalgebra of $A$. Suppose there exists some unit vector $\xi \in H$, such that (i) $A_0 \xi$ is ...
- 1,586
9
votes
2
answers
338
views
Does $End(V)$ remember $V$, where $V$ is a locally convex space?
Let $V$ be a locally convex topological vector space over $\mathbb C$, and let $A=\mathrm{End}(V)$ be its algebra of continuous linear endomorphisms (viewed just as a $\mathbb{C}$-algebra, not as a ...
- 43.2k
9
votes
1
answer
682
views
A differential inequality needed to prove a theorem about odd-dimensional souls
I need a solution to this problem (which is really a calculus problem) in order to prove a rigidity result for open nonnegatively curved manifolds with odd-dimensional souls:
Suppose that $f,g:\...
- 93
9
votes
2
answers
1k
views
Which Banach algebras are group algebras?
Given a locally compact Hausdorff group $G$, one can construct several Banach star-algebras using $G$ (and its associated Haar measure): $L^1 (G)$, $M(G)$ (regular complex measures on $G$), $L^{\infty}...
- 4,874
9
votes
2
answers
1k
views
Borsuk pairs of Banach spaces
Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$
is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$
$X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$
$\in$ $X$ ;...
- 4,060
9
votes
5
answers
870
views
Abelianization of GL(H)
This is related to Theo's question about the abelianizations of finite dimensionsal Lie groups.
I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-...
- 629
9
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1
answer
953
views
Is there always a complete, orthogonal set of unitary matrices?
The set of size-$n$ unitary matrices span $\Bbb C^{n \times n}$ (this can be proven nicely using polar decomposition). If we select a maximal linear subset of unitary matrices, then we have a basis ...
- 356
9
votes
1
answer
777
views
Abstract result on partitions of unity?
A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the ...
- 2,479
9
votes
1
answer
557
views
Dimension of eigenspaces of Laplacian on a compact Riemannian manifold
Let $M$ be a compact smooth manifold, let $g$ a riemannian metric and let $\Delta_{g}$ the Laplacian operator on functions induced by $g$. Is there a (topological?) bound on the dimension of $n$-th ...
- 91
9
votes
2
answers
928
views
Property (T) for pseudogroups
Let $H$ be a Hilbert space, $S(H)$ be the inverse semigroup (pseudogroup) of linear maps between (closed) subspaces of $H$ preserving the dot product (the operation is composition of partial maps). ...
user6976
9
votes
3
answers
3k
views
Is the "closedness of the image of operator" needed in the defintion of Fredholm operators?
in the Higson and Roe's book "analytic K-homology" just after the definition of the Fredholm operator there is a remark (2.1.3 you can see at it onlin at Google books (click here)) which claims that ...
- 91
9
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3
answers
763
views
Approximating with translated Gaussians and low-frequency trig functions
Defining the translated Gaussians by $f_t(x)=\exp(-(x-t)^2)$ for $t,x\in\Bbb{R}$, we showed that the linear span of $\{f_t \mid 0 \le t < \epsilon\}$ is dense in $L^2(\Bbb{R})$, for any $\epsilon&...
- 524
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2
answers
306
views
Ideal characterization of almost convergence
$\bullet$ A real sequence $x=(x_n)_n$ is called convergent to $\alpha$ in usual sense if for any $\epsilon>0$ the set $\{n\in\mathbb N:|x_n-\alpha|\geq\epsilon\}$ is finite.
$\bullet$ A real ...
- 930
9
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1
answer
912
views
The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?
I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a ...
- 5,169
9
votes
2
answers
321
views
For which $f \in L^2([0,1])$ is $f^\perp \cap C^\infty$ dense in $f^\perp$?
Given $f \in L^2([0,1])$, $f \neq 0$, we can consider the orthogonal complement $f^\perp$ . The smooth functions $C^\infty([0,1])$ are dense in $L^2([0,1])$. Is the intersection $f^\perp \cap C^\infty(...
- 93
9
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3
answers
2k
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Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$
Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
user47808
9
votes
1
answer
429
views
Is $\mathcal{S}(\mathbb{R}^n)$ a tame Fréchet space?
Hamilton's paper "The Inverse Function theorem of Nash and Moser" (1982, Bull. Amer. Math. Soc, vol. 7, n. 1, page $137$) proves that $C^{\infty}(M)$ is a tame Fréchet space when $M$ is a compact ...
- 370
9
votes
6
answers
839
views
Open mapping theorem for complete non-metrizable spaces?
The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...
9
votes
1
answer
1k
views
Counterintuitive consequences of the Hahn-Banach theorem
The axiom of choice has many counterintuitive consequences like the Banach-Tarski paradox.
The Hahn-Banach theorem is a consequence of the axiom of choice, but it is weaker.
I would like to know ...
- 4,461
9
votes
1
answer
401
views
Horn's inequalities for n matrices
Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$
i.e. Horn's inequalities for n matrices?
Can such ...
- 1,318
9
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1
answer
891
views
Is the space of rapidly decreasing (non-smooth) functions nuclear?
We denote by $\mathcal{S}(\mathbb{R})$ the space of smooth and rapidly decreasing functions. We define on $\mathcal{S}(\mathbb{R})$ the family of semi-norms
$$\lVert \varphi \lVert_{n,m} = \lVert (1+|...
- 2,306
9
votes
3
answers
868
views
Rosenthal like inequality for weak $\mathbb L^p$-norms
Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if $(X_i)_{i=...
- 3,958
9
votes
1
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551
views
Is the unit sphere of a Banach space dense in the unit sphere of its second dual with respect to the weak-$\ast$ topology
To be a bit more precise and fix notations, let $X$ be a Banach space (over $\mathbb{R}$ or $\mathbb{C}$), $X^{\ast\ast}$ its second dual (as a Banach space). Here and in the following we identify $X$ ...
- 1,586
9
votes
2
answers
489
views
Traces of operators in nuclear spaces
I am currently reading up on nuclear spaces in Jarchow, "Locally Convex Spaces", but I got confused and don't seem to find my mistake. In said book, theorem 21.5.9 states:
Let $F$ be a nuclear ...
- 91
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1
answer
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What are some characterizations of the strong and total variation convergence topologies on measures?
I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.
The Wikipedia article on convergence of measures defines three kinds of convergence: ...
- 203
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3
answers
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Generalizations and relative applications of Fekete's subadditive lemma
Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
- 10.5k
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3
answers
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Trace theorem for $C^{k,1}$ domains
What are the best results on (Sobolev space) trace theorems for $C^{k,1}$ domains?
For $k=0$, e.g., when the domain is Lipschitz, from e.g. the works of Martin Costabel and Zhonghai Ding, it is known ...
- 3,322
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1
answer
851
views
Banach space with uncountable basis
We know that an infinite dimensional Banach space has an uncountable Hamel basis. Now if $X$ is a vector space with an uncountable Hamel basis, does there exist a norm on $X$ for which $X$ is a Banach ...
- 585
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1
answer
333
views
Closedness of linear image of positive L1 functions
Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\...
- 537
9
votes
1
answer
608
views
Interpolation theory and $C^k$-spaces
Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
- 779
9
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1
answer
652
views
Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 124
9
votes
1
answer
609
views
Can we characterize a periodic function by the compactness of the set of its translates?
Given a function $f$, let us define the translates $f_t(x)=f(x-t)$. A (Bochner) almost-periodic function is a bounded continuous function on $\mathbb R^\nu$ such that the set of functions $\{f_t\vert ...
- 785
9
votes
1
answer
2k
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Density of smooth functions on Hölder spaces
The following result is often cited without reference in the context of PDEs:
Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
- 727
9
votes
3
answers
684
views
Lipschitz-free spaces of $\mathbb R^n$
We define
$$
\text{Lip}_0(\mathbb R^n)=\{f:\mathbb R^n\rightarrow \mathbb R, \text{such that $f(0)=0$ and }
\sup_{x\not=y}\frac{\vert f(x)-f(y)\vert}{\vert x-y\vert}<+\infty.
\}
$$
It is well-known ...
- 16.2k
9
votes
2
answers
485
views
why is this a sufficient condition for a domain to be a core of an unbounded operator?
Let $\alpha:\mathbb R\to U(H)$ be a strongly continuous action of the reals on some Hilbert space, and let $A=-i\frac d{dt}\alpha(t)|_{t=0}$ be its infinitesimal generator, so that $\alpha(t)=e^{itA}$....
- 43.2k
9
votes
2
answers
524
views
Projections onto $n$-codimensional subspaces of a Banach space: norms.
Hello, I'd like some help to find an answer I've been looking for since this morning.
Let $X$ be a Banach space and let $Y$ be an $n$-codimensional subspace of $X$. Let $P$ be a projection from $X$ ...
- 93
9
votes
2
answers
1k
views
polynomials with minimal $L_\infty$ norm on multiple disjoint intervals
It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...
- 223