Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,304
questions
16
votes
1
answer
506
views
Balls in Hilbert space
I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...
3
votes
2
answers
556
views
When is the periodisation of a function continuous?
Consider a function $f\in\mathcal{C}_0(\mathbb{R})$, where $\mathcal{C}_0(\mathbb{R})$ denotes the space of bounded continuous functions vanishing at infinity. I am interested in the $T$-periodisation ...
3
votes
0
answers
250
views
Eigenvalue bounds and triple (and quadruple, etc.) products
Very basic and somewhat open-ended question:
Let $A$ be a symmetric operator on functions $f:X\to \mathbb{R}$, where $X$ is a finite
set. Assume that the $L^2\to L^2$ norm of $A$ is $\leq \epsilon$, i....
19
votes
6
answers
8k
views
Unbounded operator bounded in a dense subset
Let $X, Y$ be normed vector spaces, where $X$ is infinite dimensional. Does there exist a linear map $T : X \rightarrow Y$ and a subset $D$ of $X$ such that $D$ is dense in $X$, $T$ is bounded in $D$ (...
2
votes
0
answers
88
views
What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?
Let $f_n: [0,1] \to \mathbb R$ be a uniformly bounded sequence in $L^p$. Then there exists a subsequence such that $f_{n_k} \to f$ weakly in $L^p([0,1])$. What is the weak limit of the sequence of ...
7
votes
3
answers
615
views
Noncommutative torus as a von Neumann algebra
Le $\theta$ be irrational. One can define the noncommutative torus $A_{\theta}$ as a universal algebra generated by two unitaries $u,v$ satisfying the relation $vu=e^{2 \pi i \theta} uv$. This is an ...
6
votes
1
answer
608
views
Closed convex hull in infinite dimensions vs. continuous convex combinations
tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$?
I am essentially asking for the most general, infinite-dimensional analogue of ...
2
votes
0
answers
41
views
Analysis of coefficients for quickly vanishing analytic vector field
Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
5
votes
1
answer
145
views
Why is density and separability needed for uniqueness of weak (time) derivatives?
Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if
$$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
5
votes
1
answer
149
views
$C^j$-topology considered by Greene and Krantz
My question is about the $C^j$ topology used by Greene and Krantz in their paper "Deformations of Complex Structures, Estimates for the $\bar{\partial}$-equation, and stability of the Bergman ...
10
votes
1
answer
509
views
discontinuous functions on the Sobolev borderline
The Sobolev embedding theorem implies that every function of class $W^{k,p}$ on a reasonable $n$-dimensional domain is continuous if $kp > n$. Cases with $kp=n$ are known as "borderline" ...
0
votes
0
answers
51
views
Double commutant theorem when $C^*$-subalgebra does not contain identity operator $1$
Double commutant theorem: For a unital $C^*$-subalgebra $M \subset B(H)$, one has
$$\smash{\overline M}^\text{SOT}=\smash{\overline M}^\text{WOT}=M''.$$
My question:
For a $C^*$-subalgebra $M \subset ...
5
votes
2
answers
220
views
Analytic approximations of smooth vector fields
Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with
$$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$
on $\mathbb{R}^3$ for any $\alpha,K$.
Further, we ...
2
votes
2
answers
240
views
inequality involving the fractional Sobolev space
Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality
$$|u(x)...
2
votes
1
answer
68
views
Equicontinuity-like property of a convex compact set
Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
Is there an ...
1
vote
1
answer
313
views
$L^p$ compactness for a sequence of functions from compactness of product with cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
3
votes
2
answers
370
views
Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different
In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}_{2}*\mathbb{Z}_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\...
2
votes
0
answers
141
views
Uniform bounds for a coupled parabolic system of PDE (linear)
Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon \...
0
votes
0
answers
45
views
Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
1
vote
0
answers
24
views
On some bounds on two constants concerning the disconnectedness of the spectra of small perturbations of operators
Let $H$ be a separable, infinite dimensional, complex Hilbert space. In the book:
Jiang, C. L.; Wang, Z. Y. (1998). Strongly Irreducible Operators on Hilbert
Space. CRC press
above the statement of ...
4
votes
0
answers
132
views
Smooth dependence of solution to elliptic pde depending on parameter
I have a question in mind but let me generalize it slightly.
Suppose I am looking at some pde like
$$-\Delta u + t u = f(u)$$ in $B_1$ (here $u=u(x)$) with $u=0$ on $ \partial B_1$ where $B_1$ is ...
4
votes
1
answer
963
views
Functional derivative of differential entropy
I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(...
1
vote
1
answer
409
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
5
votes
1
answer
343
views
Extension of functions from geodesically convex compact sets in a Riemannian manifold
In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...
1
vote
1
answer
227
views
Does the Skorokhod space with the uniform topology admit a smooth partition of unity?
Does the Skorokhod Banach space $D[0,1]$ (cadlag functions equipped with the uniform norm) admit a smooth partition of unity? I found Johanis - Smooth partitions of unity on Banach spaces, which ...
3
votes
0
answers
72
views
A holomorphic shrinking of a domain into a compact subset
This question is related to these two.
Let $X\subset \mathbb{C}^{n}$ be a bounded domain. I am interested in the following property: there is a sequence of continuous maps $\varphi_n:\overline{X}\to X$...
2
votes
0
answers
114
views
Is $A$ is small on bounded functions, is there a large subdomain on which $A$ is small?
Let $A$ be a symmetric linear operator of norm $\leq 1$ on the space of functions $f:S\to \mathbb{R}$, where $S$ is a set with $N$ elements. Define the inner product $\langle \cdot,\cdot\rangle$ by ...
7
votes
0
answers
104
views
Potential p-norm on tuples of positive operators
This is a follow-up to this question on p-norms of tuples of operators.
Consider $\left[\begin{matrix} A \\ B \end{matrix}\right] \in B(H)^2_+$, meaning $A,B\geq 0$, and define
$$
\left\|\left[\begin{...
2
votes
0
answers
62
views
Integral convergence with two sequences of functions
I came across this theorem just stated but has not proved and marked by 'it is easy to see'.
Theorem If $u_m$ and $v_m$ converges to $u$ and $v$ in $L^2([0,T];H^1(\Omega))$ weakly and $L^2([0,T];L^2(\...
2
votes
0
answers
60
views
Uniqueness of solution to Cauchy problem with quadratic nonlinearity
Consider the non-linear differential operator
$$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$
For $U\subset\...
1
vote
1
answer
205
views
A question of uniqueness
Let $u$ an harmonique function on $\Omega=(a,b)\times (0,+\infty)$ and boundary conditions :
$\displaystyle u(a,y)=u(b,y)=0,\quad\forall y\geq 0$
$\displaystyle u(x,0)=0,\,\lim_{y\to +\infty} u(x,y)=...
1
vote
0
answers
165
views
Eigenvalues of product of operators
Let $A,B$ be two Trace class operators with spectral decomposition $\sum_{j\geq 1} \lambda_j \phi_j(\cdot)\otimes \phi_j(\cdot)$ and $\sum_{j\geq 1} \gamma_j \psi_j(\cdot)\otimes \psi_j(\cdot)$ ...
5
votes
1
answer
409
views
improved Sobolev embedding
This is probably not a research level question but I am struggling with the geometry. My question is related to whether some monotonicity can increase the range of exponents in the Sobolev embedding.
...
8
votes
5
answers
477
views
Reference for : a Fréchet nuclear space is Montel
I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact"
Thank you in advance for the help!
1
vote
1
answer
753
views
Known dense subset of Schwartz-like space and $C_c^{\infty}$?
After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
3
votes
1
answer
136
views
Banach embedding of finite dimensional spaces
Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]).
I am wondering whether are any result for the case when $r>s>2$?
...
1
vote
0
answers
111
views
Algorithm/iterative procedure for constructing hypercyclic vectors?
Let $B$ be a separable Banach space and let $L:B\rightarrow B$ be a hypercyclic operator; here I use the definition of hypercyclicity given implicitly by Birkhoff's Transitivity Theorem: continuous ...
0
votes
1
answer
75
views
Ultrabornological representation for the space of uniformly continuous functions?
Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space
$$
C_{\omega_i}(\mathbb{R}^n,\...
6
votes
1
answer
216
views
Potential p-norm on tuples of operators
Consider $\left[\begin{matrix}A \\ B\end{matrix}\right] \in B(H)^2$. One can define
$$
\left\|\left[\begin{matrix}A \\ B\end{matrix}\right]\right\|_p = \| |A|^p + |B|^p\|^{1/p}.
$$
Q: Is this a norm?
...
1
vote
3
answers
430
views
Openness of the set of injective functions in $C(\mathbb{R})$?
Let $C(\mathbb{R})$ be equipped with the topology of compact convergence (or equivalently the compact-open topology). Then, is the subset $\left\{f\in C(\mathbb{R}):
\text{$f$ injective}
\right\}$ ...
3
votes
1
answer
377
views
The sign of the tail of Fourier transform of a positive function/ characteristic function
I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c_\...
18
votes
2
answers
752
views
What is known about the "unitary group" of a rigged Hilbert space?
Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ ...
4
votes
1
answer
274
views
Supremum over which sets makes $H^{\infty}$ non-separable?
It is known that the space $H^{\infty}$ of bounded holomorphic functions on the unit disk $D$ is non-separable with respect to the supremum norm $\|\cdot\|_{\infty}^{D}$. Let $E\subset D$ be connected ...
11
votes
0
answers
625
views
What is the asymptotics of the Fourier transform of $\exp(-x^4)$ for large wave numbers?
The Fourier transform of $\exp(-x^4)$ has an analytical expression, it's the difference of two generalized hypergeometric functions:
$\int d x \ e^{-x^4} e^{ikx} = 2 \ \Gamma(\frac{5}{4}) \ _0F_2(;\...
15
votes
2
answers
2k
views
In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?
This is a cross-posted on MSE here.
Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
4
votes
0
answers
113
views
If $f:U_1\to\mathcal L^p(\mu;E_2)$ is Fréchet differentiable, can we say anything about the Fréchet differentiability of $u\mapsto f(u)(\omega)$?
Let $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space, $p\ge1$, $E_i$ be a $\mathbb R$-Banach space, $U_1\subseteq E_1$ be open and $f:U_1\to L$ be Fréchet differentiable at $x\in U_1$, ...
0
votes
0
answers
83
views
Differential equation
Consider $u = u(\phi,\psi)$ where $\phi = \phi(x)$ and $\psi =\psi(x)$ are both analytic function. The following equation
$$\partial_x u - u\partial_x (\phi-\psi)=0$$
has a trivial solution $u(\phi,\...
6
votes
1
answer
465
views
Path integral as quantum mechanics on the tangent bundle
Let $X$ be a configuration space, a finite-dimensional manifold. By "quantum mechanics on $X$" I mean a linear evolution equation on complex-valued functions on $X$, determined by a ...
2
votes
0
answers
65
views
Sum of an operator with disconnected spectrum and a compact operator is strongly reducible
Let $H$ be a complex, infinite-dimensional, separable Hilbert space. Let $T \in B(H)$ be an operator with disconnected spectrum. In the introduction of the paper:
Jiang, C.; Sun, S.; Wang, Z. (1997). ...
0
votes
0
answers
99
views
Norm equivalences for Gaussian random functions (Cameron-Martin space)
Preliminaries
Consider the Hilbert space $H :={L^2_{\text{per}}(\mathbb{R})}$ of Gaussian random functions, $2\pi$-periodic in $\mathbb{R}$.
These random functions are drawn from a Gaussian measure $\...