Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,353
questions
0
votes
1
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376
views
How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?
Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
2
votes
2
answers
151
views
Can we say that : $ (A-B)\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $
Let $X$ be a separable Banach space and $A,B$ are closed convex subsets of $X$ such that $B\subset A$ and
$$
A\cap\overline{B}(0,r) \text{ and } B\cap\overline{B}(0,r) \text{ are weakly compact, } \...
9
votes
1
answer
384
views
Is $\mathcal{S}(\mathbb{R}^n)$ a tame Fréchet space?
Hamilton's paper "The Inverse Function theorem of Nash and Moser" (1982, Bull. Amer. Math. Soc, vol. 7, n. 1, page $137$) proves that $C^{\infty}(M)$ is a tame Fréchet space when $M$ is a compact ...
7
votes
0
answers
122
views
The bidual of the space of divergence-free vector fields
Consider the Banach space $L_1(\mathbb R^n, \mathbb R^n)$ of integrable vector fields $(n>1$) together with its subspace $N$ formed by those vectors fields whose divergence (computed in the ...
-2
votes
1
answer
79
views
Density property for Sobolev spaces
My question is as follows: is the space $ C_c^{\infty}(\mathbb{R}^3 \setminus \mathcal{C}) $ dense in $ H^1( \mathbb{R}^3) $ where $ \mathcal{C} $ is the circle $ \{(x,y,z) \in \mathbb{R}^3 \mid x^2 +...
2
votes
1
answer
269
views
complemented subspace of the direct sum of two Banach spaces
When I was reading a paper, I saw something like:
If $F$ and $E$ are Banach spaces with symmetric bases (precisely, they are symmetric sequence spaces), and $F$ is isomorphic to a complemented ...
2
votes
1
answer
160
views
An interpolation inequality
I am interested in the following statement. Let $q>p$. Then there are positive numbers $\alpha$ and $\beta$ so that for all $f\in C^1(\mathbb{R}^n)$, one has
$$ \left(\int|\nabla f|^p dx\right)^\...
3
votes
0
answers
204
views
Sets of finite perimeter: intersection with an half space
I have a question regarding sets of finite perimeter. In particular I'm interested to find
$$\mu_{E \cap H_t}, \label{1}\tag{1}$$
where $E$ is a set of finite perimeter in a generic open set $\Omega \...
5
votes
2
answers
351
views
Another reference for higher order Fourier analysis
I am trying to read Tao's Higher order Fourier analysis but I would be very happy to find another book on the subject. I would like to learn something about the Gowers norm and about Roth's theorem (...
3
votes
1
answer
281
views
Continuity/Lipschitz regularity of exponential map from $C_c$ to $\operatorname{Diff}_c$?
For finite-dimensional Lie algebras, see this for a nice example, the exponential map is smooth and in particular, it is locally-Lipschitz onto its image. However, things are different when moving to ...
0
votes
0
answers
65
views
Closed graph theorem for cones?
In the paper "A strong open mapping theorem for surjections from cones onto Banach spaces, Marcel de Jeu and Miek Masserschmidt, Adv. Math." it is proved (among other things) that if $X, Y$ are ...
4
votes
1
answer
188
views
Analyticity of the semigroup generated by a time-changed Brownian motion
Let $d$ be an integer. We denote by $m$ the Lebesgue measure on $\mathbb{R}^d$. We define $BL(\mathbb{R}^d)$ by
\begin{align*}
BL(\mathbb{R}^d)=\{f \in L^2_{\rm loc}(\mathbb{R}^d,m) \mid |\nabla f|\in ...
0
votes
0
answers
245
views
Images of measurable function
My question is as follows. Consider an $L^\infty$ function $f:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$ such that, for almost all $y$, the function $f({\cdot}, y)$ is continuous.
$\...
5
votes
2
answers
731
views
Injective continuous operators between Banach spaces
Suppose $X$ and $Y$ are two infinite dimensional Banach spaces. What can we say about the set of all injective continuous linear operators between $X$ and $Y$? Is it always nonempty?
4
votes
1
answer
228
views
Max-norm projection of a Hermitian matrix onto the set of positive semidefinite matrices
For a given Hermitian matrix $A$ (i.e. complex matrix with $A_{ij}^{\ast}=A_{ji}$) find its max-norm projection onto the set of complex positive semi-definite matrices:
$$\Pi(A)=\mathrm{argmin}_{M\...
-1
votes
1
answer
228
views
Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators
Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
1
vote
0
answers
72
views
"Constructive" proof that compact sets $K\subseteq L_1(\mu\times\nu)$ are contained in products $S\widehat{\otimes} T$
A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality
$$
L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu)
$$
for measures $\mu$ and $\nu$. At the same ...
1
vote
0
answers
429
views
Inner product between gradients of two functions [closed]
It is well known that the orthogonality of two functions can be measured by their inner product. check https://en.wikipedia.org/wiki/Orthogonal_functions In discrete case, the inner product is ...
0
votes
0
answers
63
views
The eigenvalue of Schrodinger opeartor
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$, consider $L=\Delta+V$ where $\Delta=div\nabla$. According to section 8.12 in Gilbarg and Trudinger's book, if $V\in L^\infty(\Omega)$, then ...
7
votes
1
answer
331
views
Prokhorov theorem on non Polish spaces
It is well known that if $X$ is a Polish space and $\mathcal{F} \subset \mathcal{M}_+(X)$ (the set of finite positive Radon measures on $X$) is uniformly tight and bounded in mass, it is relatively ...
-2
votes
1
answer
967
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
1
vote
0
answers
145
views
Spectrum of Laplacian-like operator
Let $\kappa_1, \kappa_2>0$ be fixed.
Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by
$$
A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
1
vote
0
answers
202
views
Construction of weight function to satisfy condition on given functional
Consider the following function :
$$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$
Here, $\omega(z)$ is a weight we are going to consider
The following two conditions should meet for $\...
3
votes
0
answers
81
views
Compatibility between the source and the boundary condition for an Helmholtz-type equation
Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
8
votes
1
answer
368
views
A question about comparison of positive self-adjoint operators
I have the following question but have no idea on its proof (one direction is trivial):
Let $A$ and $B$ be (bounded) positive self-adjoint operators on a complex Hilbert space $H$. Prove that
$$\...
1
vote
1
answer
165
views
Integrable functions as elements of closed absolutely convex hulls of precompact sets of indicator functions
I am not a specialist in measure theory, so excuse me if this is simple.
Let $\mu$ be a finite measure on a set $X$ (for example, the Lebesgue measure on $[0,1]$). Integrable functions on $X$ can be ...
1
vote
1
answer
396
views
Convergence in $C_c$ but not in $C$
Let $C_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ ...
2
votes
1
answer
148
views
Ergodic decomposition - how does restricting measure effect it? (Choquet Theory)
Suppose that $G$ is a discrete countable group and $\mu$ is an IRS (invariant random subgroup) of $G$: $\mu$ is conjugation invariant as a probability measure on the subgroups of $G$.
Since all the $...
3
votes
2
answers
646
views
Gradient flows: convex potential vs. contractive flow?
Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$).
Consider the autonomous gradient-flow
$$
\dot ...
2
votes
3
answers
306
views
What are the necessary conditions on $f$ if $f(x)=f(\sin(\pi x)+x)\iff x\in\Bbb{Z}$?
I am aware that the statement:
$$f(x)=f(\sin(\pi x)+x)\iff x\in\Bbb{Z}$$
is not true for all $f$. For example, $f$ can be $x$ to any constant power or any constant to the $x$th power but it cannot be ...
9
votes
1
answer
606
views
Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity
In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...
8
votes
1
answer
153
views
closed ideals in L(L_1)
Denote $L_1=L_1[0,1]$ The lattice of closed ideals in $\mathcal{L}(L_1)$ includes the chain
$$
\{0\}\subsetneq\mathcal{K}(L_1)\subsetneq\mathcal{FS}(L_1)
\subsetneq\mathcal{J}_{\ell_1}(L_1)\subsetneq\...
3
votes
1
answer
142
views
Density of $C(X,\operatorname{co}\{\delta_y\}_{y \in Y})$ in $C(X,\mathcal{P}(Y))$
Let $X,Y$ be locally-compact Polish spaces, equip the set $\mathcal{P}(Y)$ of probability measures on $Y$ with the weak$^{\star}$ topology (topology of convergence in distribution), and equip $C(X,\...
1
vote
0
answers
38
views
$C^2$-control using orthonormal frame on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold. Let $E=(E_1,\dots,E_n)$ be an orthonormal frame for $M$.
So for $M$ itself we have a natural $C^k$-norm $\|f\|_{C^k_g(M)}:=\max\limits_{1\le m\le k}\sup\limits_{...
1
vote
0
answers
102
views
Is this a positive definite kernel?
Under which conditions on the function :
\begin{array}{l|rcl}
K : & \mathbb R^+ & \longrightarrow & (0, 1)\\
&t & \longmapsto & K(t) \end{array}
is the symmetric ...
4
votes
0
answers
184
views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
-1
votes
2
answers
468
views
inequivalent norms [closed]
I am thinking about the following question:
Let $X$ be a Banach space, say separable, e.g., $l_p$ or $c_0$.
When can I say that there exist inequivalent complete norms on $X$?
3
votes
2
answers
492
views
if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?
This question is related to solving $f(f(x))=g(x)$.
Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...
1
vote
0
answers
67
views
Comparison of inductive limit topology with very-rapidly decaying non-convex $L^{!/2}$-space topology
This is related to these posts and here.
Let $L^1([n,n+1])$ denote the subspace of $L^p$-functions on $[0,\infty)$ essentially supported on $[-n,n]$. Denote the accelerated $\ell^1$-direct sum ...
0
votes
0
answers
151
views
Use of this space of very rapidly decreasing continuous functions
Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space
$$
V_p
:=
\left\{
f \in C([0,\infty)):\,
\sum_{n=1}^{\infty} ...
0
votes
1
answer
155
views
Showing a product on a character space is continuous
Quoting from Timmermann's An invitation to quantum groups and duality:
Prop. 5.1.3 Let $A$ be a commutative algebra of functions on a compact
quantum group. Then there exists a compact group $G$ and ...
4
votes
0
answers
305
views
MOOC's courses on Functional analysis [closed]
Several years ago, Coursera offered a MOOC (Massive Online Open Course) on Functional Analysis:
An Introduction to Functional Analysis. Taught by John Cagnol.
Ecole Centrale Paris.
The duration was ...
0
votes
0
answers
131
views
Spectral Theorem for compact self-adjoint operators on real Hilbert spaces [duplicate]
Is the spectral theorem for self-adjoint compact operators on a Hilbert space also true if the Hilbert space is real (instead of complex)?
Wikipedia says this is true.
However, it seems to me that ...
7
votes
1
answer
385
views
Why are we interested in operators that share a basis of eigenfunctions?
I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.
I ...
4
votes
1
answer
178
views
Continuity upgrade for nonlinear maps
Let $E,F,G$ be topological vector spaces such that $F\subset G$ with continuous embedding.
By continuity upgrade I mean the following phenomenon: In some circumstances a continuous linear map $f:E\...
2
votes
2
answers
395
views
Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$
Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
5
votes
1
answer
360
views
Inverse marginal property of a collection of $\sigma$-algebras
In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space"
I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras.
Let $(\...
2
votes
1
answer
510
views
Separable Banach spaces isometric to quotient of a Banach space
We know that every separable Banach space is isometrically isomorphic to a quotient space of $(\ell^1,\|.\|_1)$. We also know that the norm defined by $\|x\|=(\|x\|_1^2+\|x\|_2^2)^{1/2}$ for all $x\in ...
2
votes
1
answer
284
views
Sum of strongly commuting self-adjoint operators
Let $A,B$ be two positive unbounded, self-adjoint operators on some Hilbert space that strongly commute. Let $D(A)$ and $D(B)$ denote their respective domain. Then, using for instance the spectral ...
3
votes
0
answers
251
views
Kazhdan Property T of semisimple Lie groups
I am reading the paper [Margulis, G. A.; Nevo, A.; Stein, E. M.,
Analogs of Wiener's ergodic theorems for semisimple Lie groups. II.
Duke Math. J. 103 (2000), no. 2, 233–259] (MSN).
I want to ...