Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
5,629 questions
0
votes
0answers
121 views
A maximization problem
Consider the minimization problem described this paper. Let $f_{\lambda}$ be the minimizer. As a part of extending my work, I am able to show the following facts
$$\lim_\limits{\lambda \to 0}\|f_{\...
5
votes
0answers
83 views
How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
1
vote
0answers
42 views
The tower of path algebras associated to a tower of finite dimensional $C^*$-algebras is isomorphic to the original tower
Let $A_0\subseteq A_1\subseteq...$ be an infinite tower of unital inclusions of finite dimensional $C^*$-algebras and $B_0\subseteq B_1\subseteq ...$ be its associated infinite tower of path algebras. ...
3
votes
1answer
132 views
Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization
The following result is well-known:
Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have
$$H \left( (1 - \lambda)(x_1,y_1) + \...
2
votes
0answers
63 views
Exp-decay estimate of Schrodinger equation
Consider the equation $Hu=0$ with $u\in L^2(\Omega)$, where $H=-\Delta+V$ for some bounded continuous function $V$ and $\Omega$ is an un-bounded domain(e.g. $\mathbb R^n$). If $0$ is in discrete ...
1
vote
0answers
79 views
Functional Analysis book with graphical descriptions [on hold]
I am looking for some book about functional analysis that has also graphical descriptions and images of the important concepts.
Anyone knows some book like this?
0
votes
0answers
36 views
Show convergence of a sequence of resolvent operators
Let
$E$ be a locally compact separable metric space
$(\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $C_0(E)$
$E_n$ be a metric space for $n\in\mathbb N$
$(\...
1
vote
0answers
22 views
Is there an easy characterisation (perhaps some generalised Löwner representation) for operator monotone functions of order $n$?
As per my understanding, roughly stated, $f$ is an operator monotone function of order $n$ if for all $n\times n$ (Hermitian) matrices, $X,Y\ge0$ which satisfy $X\ge Y$, we have $f(X)\ge f(Y)$.
If $f$...
2
votes
0answers
61 views
Gagliardo-Nirenberg inequality for periodic functions?
I am interested in Gagliardo-Nirenberg type inequality (see https://en.m.wikipedia.org/wiki/Gagliardo%E2%80%93Nirenberg_interpolation_inequality) for functions in the space
$$H^1_T(\mathbb{R}^n)=\...
6
votes
4answers
199 views
Open mapping theorem for complete non-metrizable spaces?
The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's ...
2
votes
0answers
32 views
Time derivative in parabolic Hölder spaces
Let $\Omega$ be a regular open set in $\mathbb{R}^n$ and $T>0$.
Let $C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$ be the space of functions $f$ which are $\frac{1+\alpha}{2}$-...
1
vote
0answers
32 views
$W^{k,p}$ and Holder regularity for linear elliptic systems with Neumann boundary data
I'm looking for a text or paper that discusses regularity in the Sobolev and Holder sense for general linear elliptic systems of PDEs on bounded domains with Neumann boundary data.
The book by ...
4
votes
1answer
126 views
Relation between two different functionals: $\lVert p^{-\max}_{-\varepsilon}\rVert$ and $\kappa_{p}^{-1}(1-\varepsilon)$
Given a non-negative sequence $p=(p_i)_{i\in\mathbb{N}}\in \ell_1$ such that $\lVert p\rVert_1 = 1$,we define the two following quantities, for every $\varepsilon \in (0,1]$.
Assuming, without loss ...
4
votes
0answers
86 views
Korovkin subset of $C(\mathbb{T})$
Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\...
5
votes
1answer
183 views
+50
Non-uniform property of sequences
Let us say a sequence $(x_n)_{n=1}^\infty$ in some Banach space $X$ has $S_C$ if there exist $k_1<k_2<\ldots$ such that for any $t\in \mathbb{N}$ and scalars $(a_n)_{n=1}^t$, $$\|\sum_{n=1}^t ...
4
votes
1answer
106 views
Comparison of several topologies for probability measures
Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
1
vote
0answers
40 views
Strong Differentiability of Spectral Projections
Let $H$ be a Hilbert space and $W$ be a dense subspace, equipped with a different norm that turns it into a Hilbert space. Let $(A(t))_{t\in[0,T]}$ be a family of Operators in $B(W,H)$ (bounded ...
5
votes
1answer
139 views
Textbook recommendations: Weakly almost periodic functions
I am currently studying Ergodic Theory from Glasner’s book - in it, weakly almost periodic functions play a large role, as well as general “means” and unitary representations of groups on Hilbert ...
0
votes
0answers
31 views
Convergence of regression coefficients to probability density
By simulation we create a vector $Y = (y_1,y_2,...,y_n)$, where each $y_i \in R$ is independently drawn from a given non-degenerate distribution.
Next we create by simulation a vector $\xi = (\xi_1,\...
0
votes
1answer
47 views
Box counting dimension of a set and Lipschitz functions
If $f$ is Lipschitz, then the following holds for the Hausdorff dimension:
$$\dim_H f(A) \le \dim_H A.$$
Is the same true for the box counting dimension?
-2
votes
0answers
41 views
Box counting dimension of the graph of the Cantor function
Consider the Cantor staircase function. The Hausdorff dimension of its graph is $1$. What is the box-counting dimension of its graph?
A more general question is Box counting dimension of the graph of ...
0
votes
0answers
47 views
An inequality for norms involving convolutions [migrated]
I need to prove/disprove the following existence theorem:
$$
\forall \epsilon > 0, \quad \exists f\in L^1(\mathbb R), g\in L^p(\mathbb R) : \quad (1-\epsilon)\|f\|_1\|g\|_p < \|f\ast g\|_p,
$$
...
1
vote
0answers
101 views
Is this integral zero?
I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation.
Consider $\mathfrak{g}=C^{\infty}(M)$ and the dual $\mathfrak{g}^*=\...
1
vote
1answer
111 views
Hausdorff dimension of the graph of the sum of two continuous functions
How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions:
Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
3
votes
1answer
216 views
+100
Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different
In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}_{2}*\mathbb{Z}_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\...
2
votes
0answers
90 views
Euler-Lagrange equations on a differentiable manifold
I am following the conventions of https://arxiv.org/abs/math-ph/9902027
Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over $...
3
votes
1answer
133 views
Uniform convergence of generalised Fourier series
Suppose $u_n$ is an orthonormal basis of smooth functions on $S^1$.
Does there exist a smooth function $u$ such that the generalised Fourier series
$$u=\sum_{n\in\mathbb{N}} \langle u,u_n\rangle u_n ...
3
votes
1answer
114 views
When does the spectral radius strictly increase?
For bounded linear operators $A$ and $B$ on a Banach space $X$, I'm looking for results which imply that $r(A) < r(A+B)$ (note the strict inequality), where $r(A)$ denotes the spectral radius of $A$...
1
vote
0answers
39 views
Derivation of the vortex filament equation from Euler equation
How can the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
be derived from the Euler equation
$$\partial_t \...
5
votes
0answers
43 views
First Cousin Problem for Bergman spaces
I recall (an easy case of) the first Cousin problem :
Let $\Omega_1, \Omega_2$ be two open subsets of the complex plane
$\mathbb{C}$ with non-empty intersection and $f$ be holomorphic on $\...
3
votes
1answer
44 views
Reference Request: $L^p(x)$/(Musielak–Orlicz space) analogue of classical $L^p$ result
Fix a non-empty open domain $\Omega\subseteq \mathbb{R}^d$ with compact closure, and a finite Borel measure $\mu$ on its closure $\overline{\Omega}$.
In Halmos' book it is shown that:
Classical ...
2
votes
1answer
99 views
Embedding of weighted sobolev space with exponential weights
In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. ...
4
votes
0answers
38 views
Commuting flows problem for non-Lipschitz vector fields
Let $X$ be a continuous vector field on a (say compact) manifold $M$, if $X$ has ODE uniqueness then we can define its associated flow $\mathcal F_X:\mathbb R\times M\to M$ uniquely given by $\mathcal ...
-1
votes
0answers
21 views
Regression with incomplete basis [closed]
Suppose that $f(x) = \frac{1}{2}\text{max}(x+10,0) + \frac{1}{3}\text{max}(x+20,0) + \frac{1}{4}\text{max}(x+110,0) + \frac{1}{5}\text{max}(x+120,0)$
If I randomly simulate values for $x$ such that $$...
3
votes
1answer
53 views
How does $E$ closed follow from the upper semicontinuity of the spectrum?
Let $f$ be an analytic function for a domain $D$ of $\mathbb{C}$ into a Banach algebra $A$. Suppose that, for all $\lambda \in D$, $\text{Sp}f(\lambda)$ is finite or a sequence converging to $0$.
...
7
votes
0answers
139 views
Does this ideal in $B(L_1)$ have a (bounded) right approximate identity?
I will take a roundabout way to defining this ideal, because (a) this route is how my collaborators and I came to it (b) this alternative definition, rather than the standard one, may suggest a ...
1
vote
0answers
48 views
Global solution of nonlinear Schrödinger equation via blow-up argument
Set $u_0\in H^1 (\mathbb{R} ^N)$ and $1<\alpha < \frac{N+2}{N-2}$.
I want to show that there exists $\varepsilon > 0$ s.t. if $\Vert u _0 \Vert _ {H^1} < \varepsilon$,
then there is ...
5
votes
0answers
71 views
Projection of a function $f\in L^1(\Omega)$ onto a finite dimensional subspace
Suppose $\Omega \subset \Bbb R^n$ be a domain such that $|\Omega|<\infty$, $f\in L^1(\Omega)$. Let $Y= \text{span}\{g_1,\dots, g_k\}$.
Is there a characterization of the set of projections of $f$...
2
votes
0answers
74 views
Question: can we claim that $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$?
Let $\Omega$ be a compact manifold in $\mathbb R^2$. For $1 \leq p \lt
4/3$ can we claim that
$W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$
with the first inclusion being ...
2
votes
0answers
75 views
Minimizing sequence $\implies$ Palais–Smale sequence
Set $F:\mathbb{R}^n\rightarrow \mathbb{R}$ a $C^2$-function that is bounded from below. Set $x_n$ a minimizing sequence, i.e., $F(x_n)\to \alpha = \inf F$. I want to prove that under the assumption of ...
2
votes
1answer
71 views
Closable unbounded operators and Banach space adjoints
For an unbounded operator $T:\mathcal{H}_1 \to \mathcal{H}_2$, if its adjoint $T^*$ is densely defined, then we know that $T$ is closable. What happens if we replace $\mathcal{H}_1$ or $\mathcal{H}_2$ ...
1
vote
0answers
25 views
Showing there exists a solution to a variational inequality
I'm working through a book that provides the following exercise which I'm having trouble with. They cite this paper which I can't find (and even if I could, I can't read italian).
The problem:
Let ...
3
votes
0answers
46 views
Is projection method really applicable for numerical solution of linear integral equations in $ L^p \ (p \neq 2)$ setting?
Projection method is a traditional method to numerically handle problem of linear integral equation. The routine way is to do it in $ L^2 $ setting. For example:
Let $ A:L^2(a,b) \to L^2(a,b) $ be a ...
4
votes
1answer
157 views
Does Banach-Mazur distance between regular polygons admit any structure that lends to approximation or exact results in particular situations?
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides. Do closed form or approximate results exist (at least at special infinitely ...
5
votes
1answer
157 views
Unbounded Fredholms operators
Motivated by the situation of bounded Fredholm operators, I have the following question about "unbounded Fredholm operators".
Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be two Hilbert spaces, and
$$
D: ...
3
votes
2answers
163 views
Interior smooth regularity
I recently read the PDE book of L. Evans, and in its chapter 6 some kinds of regularities of second-order elliptic equations were discussed. My question is about its proof of interior smooth ...
1
vote
0answers
30 views
A linear first order PDE with boundary condition
I want to solve the following first order PDE
$$
(\star)\quad\begin{cases}
\nabla u\cdot \nabla\xi=f \quad\text{in}\,\Omega, \\
u\mid_{\partial \Omega}=0
\end{cases}
$$
where $\xi\in C^2(\overline{\...
0
votes
1answer
86 views
Reference request : How to use Lagrange multiplier technique with infinite (infact uncountably) number of constraints?
I have a constrained maximization problem (maximizing a functional), with number of constraints being uncountable infinite.
It looks something like this. I want to maximize the convex functional $C(f)...
2
votes
3answers
149 views
Every linear topological space embeds into the Tychonoff product of linear metric spaces
I need a reference to the following (known?)
Fact. Every topological vector space $X$ over the field of real numbers is topologically isomorphic to a linear subspace of the Tychonoff product of ...
4
votes
1answer
126 views
Compute $ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE
Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator.
How can I compute the ...
