Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,304
questions
19
votes
2
answers
6k
views
Question about functional derivatives
This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
-1
votes
1
answer
117
views
Existence of a function with slow growth on derivatives
Does there exist a smooth compactly supported function $$f \in C^{\infty}_c((0,1))$$
such that
$$ \|D^k f\|_{L^{2}(0,1)} \leq \left\lfloor{\alpha\,k}\right \rfloor! \quad \forall\, k\in \mathbb N$$
...
2
votes
1
answer
167
views
Relation between Gevrey space $G^s$ and Schwartz space $S$
I want to decide whether there is a relation between the Gevrey space $G^s(\mathbb{R}^m)\;(s>1)$ and the schwartz space of rapidly decreasing functions $S(\mathbb{R}^m)$. I know that polynomials ...
1
vote
0
answers
87
views
Explanation for the energy method used here
I am reading a paper where the authors prove
$$ \frac{d}{dt} \Vert f \Vert_{H^4} (t)\leq C\Vert f \Vert^5_{H^4}(t) $$
Where $f=f(x,t)$ and $H^4$ stands for the usual Sobolev space. Using Gronwall's ...
3
votes
3
answers
531
views
Did anyone ever introduce an "oscillating unity"?
I wonder whether anyone ever tried to introduce an extension of real numbers by adding an element $\nu$ which would signify the behavior of the function $(-1)^x$ as $x$ goes to infinity?
In other ...
0
votes
0
answers
175
views
Canonical embedding of Hilbert space in random $L^2$
This question is a followup of Canonical embedding of Hilbert space in $L^2$ space, where it was essentially shown that there is no canonical way to construct, from an abstract Hilbert space $H$, a ...
1
vote
1
answer
113
views
$K(x,y)\in L^{\infty}(R^n\times R^n, m\times m)$, $K(x,y)=K(y,x)$, so $K(x,y)=\sum_{k=1}^{\infty}\lambda_k \phi_k(x)\phi_k(y)$, are $\phi_k$ bounded?
Consider a symmetric function
$$
K(x,y):R^n \times R^n \to R
$$
satisfying $K(x,y)=K(y,x)$ and
$$
\int_{R^n\times R^n} K^2(x,y)dm(x) dm(y) <\infty.
$$
Let $m$ be a probability measure on $R^n$.
...
3
votes
1
answer
145
views
spectrum of multiplicative morphisms
Let $T:[0,1]\to[0,1]$ be a continuous map, which is neither surjective nor injective. Put
$$
C([0,1])\ni f\mapsto \Phi(f):=f\circ T\in C([0,1]).
$$
Notice that, under the above conditions, $0\in\sigma(...
2
votes
2
answers
179
views
MP critical point has morse index 1, proof
I wonder where can I find a proof of the following fact: if the mountain pass critical point is non-degenerate, then its Morse index is 1. I am very interested in reading it.
In general, I am ...
2
votes
1
answer
241
views
Estimate the metric entropy of unit ball in $L^2$ space
Let me clarify the setting I'm thinking.
For any totally bounded metric space $(Y,d_Y)$ and $\varepsilon>0$, the $\textit{metric entropy}$ $N_M(\varepsilon,Y)$ is the smallest number of closed ...
4
votes
0
answers
143
views
A Pythagorian inequality characterization of inner-product spaces
Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\...
3
votes
1
answer
232
views
Regularity and normal trace of "Hdiv" measures
In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$.
I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with ...
1
vote
1
answer
151
views
If $f(x_1,x_2)=f(x_2,x_1)$, $f(x_1,x_2)=\sum_k \lambda_k f_k(x_1)f_k(x_2)$? [closed]
Consider a symmetric function
$$
f(x_1,x_2):R^n \times R^n \to R
$$
satisying $f(x_1,x_2)=f(x_2,x_1)$. Are there functions $f_k:R^n \to R$ such that
$$
\int_{x\in R^n}f_k(x)f_l(x)dm=\delta_{kl},
$$
...
1
vote
0
answers
88
views
Do we have $M\hat{\otimes}_A N\cong M\otimes_A N$ if $M$ is a finitely generated projective $A$-module over a nuclear Frechet algebra $A$?
Let $A$ be a nuclear Frechet algebra with unit. Let $M$ be a right Frechet $A$-module and $N$ be a left Frechet $A$-module. Both $M$ and $N$ are assumed to be non-degenerate. We can define the ...
0
votes
1
answer
394
views
Canonical embedding of Hilbert space in $L^2$ space
Let $H$ be a Hilbert space. I am interested in isometries $f\colon H\to L^2(X,\mu)$ where $\mu$ is a probability measure on some measure space $X=(X,\mathcal F)$ where $\mathcal F$ is a $\sigma$-...
0
votes
0
answers
83
views
Prove that the solution belong to ${L^2}\left( {0,T;{L^2}\left( \Omega \right)} \right)$
Let $k \in {L^\infty }\left( {0,T} \right)$ and we assume that
$$\phi :t \mapsto u\left( t \right) + \int_0^t {k\left( s \right)u\left( s \right)ds} \in {L^\infty }\left( {0,T;{L^2}\left( \Omega \...
3
votes
1
answer
331
views
When does $\lVert T_n \rVert_p \to \lVert T_0 \rVert_p$ imply $\lVert T_n - T_0 \rVert_p \to 0$?
maybe this question is too elementary but I could not find any results in the functional analysis textbooks I own:
For separable Hilbert spaces $H$, $K$, let $S_p(H,K)$ be the $p$th Schatten class of ...
2
votes
0
answers
225
views
Links between differing notions of "pseudo-measure"'; or, why that name?
(A pet peeve of mine is Mathematicians from field X noticing that field Y uses terminology which is very close to that from field X, and assuming there are Mathematical links. This question might be ...
-1
votes
1
answer
80
views
Closed on generic set implies closed set whole set [closed]
Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense ...
4
votes
1
answer
336
views
Removing the interior of spectrums
Let $A$ be a Banach algebra. Is there a Banach algebra $B$ which contains $A$ but the spectrum of each elements of $B$ has empty interior(as a subset of $\mathbb{C}$)?
The motivation comes from the ...
2
votes
1
answer
174
views
The inclusion $G^s\subset C^\infty$ is strict [closed]
I'm studying the Gevrey class $G^s,\;s>1$, which is a subset of the $C^\infty$ class. I want to find an example of a function that is $C^\infty$ but not $G^s$.
14
votes
3
answers
741
views
Is $C^{\infty}(\mathbb{R}^{m+n})$ a flat module over $C^{\infty}(\mathbb{R}^{m})$?
For $m>0$ we consider the ring $C^{\infty}(\mathbb{R}^{m})$ of smooth functions on $\mathbb{R}^{m}$. For $n>0$ we consider the projection $\mathbb{R}^{m+n}\to \mathbb{R}^{m}$ hence $C^{\infty}(\...
6
votes
0
answers
132
views
Does every locally convex space with a Schauder basis have the approximation property?
For Banach spaces, the existence of a Schauder basis implies that this space has the approximation property.
Since both the notion of Schauder bases and of the approximation property are well ...
2
votes
0
answers
166
views
A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
2
votes
2
answers
254
views
Do we have a name for this space?
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Consider the class
$$
\mathcal{F}=\{f\in L^{1}(\Omega):\exists C>0 \text{ s.t. } \int_{U}|f|\leq C\sqrt{|U|},\text{ for any }U\subset \Omega.\...
28
votes
6
answers
5k
views
Any real contribution of functional analysis to quantum theory as a branch of physics?
In the last paragraph of this last paper of Klaas Landsman, you can read:
Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...
0
votes
1
answer
158
views
Bisector Projection
Let $p,q$ be two projections of a $C^*$ algebra. A projection $l$ is called a bisector projection to $(p,q)$ if $$|pl-l|=|ql-l|$$ The motivation comes from the geometric intuition of "...
3
votes
0
answers
81
views
Reference request for representation theory of TRO
Let $H$ and $K$ be Hilbert spaces. Recall that a Ternary ring of operator(TRO) $V$ is a closed subspace of $B(H,K)$ such that $xy^{\ast}z \in V$ for all $x,y,z \in V$. I have recently started reading ...
4
votes
0
answers
124
views
Relationship between three different definitions of solutions for ODE with irregular coefficient
What is the difference between the notions of
Regular Lagrangian flow
Filippov solution
Caratheodory solution
of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
1
vote
1
answer
105
views
A unitary matrix of functions [closed]
If $A(z)=[A_{ij} (z)] $ is an $n\times n$ unitary matrix valued functions. Is there a characterization of such matrix if:
(1) the entries are analytic functions on a set $D$.
and
(2) if the ...
4
votes
1
answer
340
views
Approximation property of a Banach space in terms of finite-rank projections
Let $X$ be a separable Banach space. Is this property equivalent to the approximation property?
There exists a chain $X_n$ of finite-dimensional subspaces of $X$, each being a range of some ...
1
vote
0
answers
290
views
Continuity of the Legendre transform of a Lipschitz function
Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...
-1
votes
1
answer
317
views
Sequence converging to different limits with respect to two different _complete_ norms
Do there exist a real vector space $X$ complete with respect to norms $|\cdot|$ and $\|\cdot\|$ and a sequence $(x_n)_{n\in \mathbb N} \subset X$ such that there exist $x,y\in X$: $x\ne y$, $|x_n - x|\...
3
votes
0
answers
250
views
How can we solve this kind of saddle point problem?
I'm trying to solve a saddle point problem of the following form: Let
$(E,\mathcal E,\lambda)$ be a measure space;
$p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$
$W$ be ...
7
votes
2
answers
495
views
Cut norm versus $l_1$ norm
Let $K$ be the set of $n\times n$ matrices with zero diagonal entries and such that the sum of all entries is zero.
The cut norm of a $n\times n$ matrix $M$ is:
$$
cut(M) = \sup_{S, T, S\cap T = \...
8
votes
2
answers
318
views
Completeness of exponentials $\mathcal{E} = \{ e^{ist} : s \in \mathbb{R} \}$ in $L^p(\mu)$
In the paper A problem on completeness of exponentials (Annals of Mathematics 178 (2013), 983-1016), the author A. Poltoratski studies the following problem:
Let $\mu$ be a finite positive measure on ...
9
votes
1
answer
881
views
Completed tensor product is exact
In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on ...
6
votes
2
answers
393
views
Subspaces of $\ell_p$ ($1<p<\infty$, $p\neq 2$) not isomorphic to $\ell_p$
Is it possible to show the existence of an infinite dimensional closed subspace of $\ell_p$ ($1<p<\infty$, $p\neq 2$), not isomorphic to $\ell_p$, in an elementary way?
For $1<p<q<2$,...
4
votes
1
answer
115
views
Notion of a "smooth function of the order two" (Yakubovich, "Index Transforms")
In the book "Index Transforms" by S.B. Yakubovich (World Scientific, 1996), I encountered the notion of a "smooth function of the order two" with compact support on $\mathbb R_+$, denoted $C^{(2)}(\...
4
votes
1
answer
135
views
Are unit balls in Banach spaces retracts of bidual balls?
Let $X$ be a separable Banach space embedded canonically in $X^{**}$. Is there a retraction from the unit ball $B_{X^{**}}$ of $X^{**}$ onto the unit ball $B_X$ of $X$?
When we insist on uniformly ...
2
votes
1
answer
93
views
Identification of some strong topology
Let $X$ be a completely regular Hausdorff space, $C_b(X)$ the Banach space of bounded continuous function and $M(X) \subseteq M(\beta X) = C(\beta X)' = C_b(X)'$ the spaces of Radon measures on $X$. ...
0
votes
1
answer
64
views
Is there a general guideline for minimizing $\sup_{y\in H}F(\;\cdot\;,y)$?
Let $H$ be a $\mathbb R$-Hilbert space and $F:H^2\to\mathbb R$. Is there a general guideline for minimizing $\sup_{y\in H}F(\;\cdot\;,y)$?
Since the question is rather abstract, feel free to impose ...
1
vote
2
answers
409
views
Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x)), &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
where $\chi$ denotes the ...
6
votes
0
answers
99
views
Is every separable Banach space with the MAP 1-complemented in a space with a monotone basis?
The question, already phrased in the title, looks like a classical problem from Banach space theory from the 1970s. Hence, my question is more of a reference request in its nature.
Can every ...
2
votes
3
answers
213
views
Equivalence of operators
let $T$ and $S$ be positive definite (thus self-adjoint) operators on a Hilbert space.
I am wondering whether we have equivalence of operators
$$ c(T+S) \le \sqrt{T^2+S^2} \le C(T+S)$$
for some ...
2
votes
0
answers
2k
views
On weak compactness of the unit ball in a reflexive Banach space
It is a well known result in functional analysis that a Banach space $X$ is reflexive if and only if the unit ball is weakly compact (compact in the weak topology). This result is also known as ...
3
votes
1
answer
360
views
Positive part of Cauchy sequence of Sobolev functions is again Cauchy
Let $p\geq 1$ and consider the space $W^{1,p}(B)$ where $B\subset \mathbb{R}^{n}$ is the standard unit ball. Moreover, let $f_{k} \in C^{\infty}(B)$ be a Cauchy sequence in $W^{1,p}(B)$ of smooth ...
23
votes
4
answers
4k
views
Most general definition of differentiation
There are various differentiations/derivatives.
For example,
Exterior derivative $df$ of a smooth function $f:M\to \mathbb{R}$
Differentiation $Tf:TM\to TN$ of a smooth function between manifolds $f:...
4
votes
2
answers
305
views
Do all unitary representations weakly converge to zero at infinity?
Question. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a Radon measure space. Let $$\rho\colon G\to U(L^2(X))$$
be a unitary, strongly continuous, representation. Is ...
3
votes
0
answers
149
views
Completeness of discrete shifts in $\mathbb{R}^n$
Consider the space $L^2(\mathbb{R})$. Let $(x_n)_n \subset \mathbb{R}$ be a sequence and $f \in L^2(\mathbb{R})$ a functions. It is well understood under which assumptions the span of the set
$$
S = \{...