Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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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 ...
IamWill's user avatar
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-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 $$ \|D^k f\|_{L^{2}(0,1)} \leq \left\lfloor{\alpha\,k}\right \rfloor! \quad \forall\, k\in \mathbb N$$ ...
Ali's user avatar
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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 ...
Jem Y's user avatar
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1 vote
0 answers
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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 ...
R. N. Marley's user avatar
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 ...
Anixx's user avatar
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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 ...
pre-kidney's user avatar
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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 $R^n$. ...
mathmetricgeometry's user avatar
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(...
fidaleo's user avatar
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2 votes
2 answers
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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 ...
R. N. Marley's user avatar
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 ...
S.Lim's user avatar
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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\...
Iosif Pinelis's user avatar
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 ...
leo monsaingeon's user avatar
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 $$ \int_{x\in R^n}f_k(x)f_l(x)dm=\delta_{kl}, $$ ...
mathmetricgeometry's user avatar
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 ...
Zhaoting Wei's user avatar
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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 $\sigma$-...
pre-kidney's user avatar
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0 answers
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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 \...
Trần Quang Minh's user avatar
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 ...
r_faszanatas's user avatar
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 ...
Matthew Daws's user avatar
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-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 ...
Adam's user avatar
  • 1,001
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 ...
Ali Taghavi's user avatar
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 $G^s$.
Jem Y's user avatar
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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}(\...
Zhaoting Wei's user avatar
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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 ...
Christian's user avatar
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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|_{\...
MathLearner's user avatar
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.\...
Ahmed Tori's user avatar
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 "...
Ali Taghavi's user avatar
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 ...
Math Lover's user avatar
  • 1,065
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 $\Phi(0,x) = x$, ...
Riku's user avatar
  • 819
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 ...
Guest's user avatar
  • 39
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 ...
A. U.'s user avatar
  • 97
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_{...
Alfred's user avatar
  • 31
-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|\...
Skeeve's user avatar
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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 ...
0xbadf00d's user avatar
  • 161
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 = \...
alesia's user avatar
  • 2,582
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 ...
Jamie Mathews's user avatar
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 ...
Boris Bilich's user avatar
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 $1<p<q<2$,...
M.González's user avatar
  • 4,301
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)}(\...
kolaka's user avatar
  • 295
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 ...
A. U.'s user avatar
  • 97
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 $X$. ...
yada's user avatar
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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 ...
0xbadf00d's user avatar
  • 161
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 ...
Riku's user avatar
  • 819
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 ...
Tomasz Kania's user avatar
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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 ...
van Dyke's user avatar
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 ...
Arian's user avatar
  • 364
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 ...
BremerH's user avatar
  • 49
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:...
Ponta's user avatar
  • 361
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 ...
Giuseppe Negro's user avatar
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 = \{...
Muzi's user avatar
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