Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,354
questions
8
votes
2
answers
532
views
Are (completely) positive maps approximated by normal (completely) positive maps?
Let $\mathcal{H}$ denote a Hilbert space and $B(\mathcal{H})$ denote the algebra of all bounded operators on $\mathcal{H}$. By recognizing the (Banach) dual of $B(\mathcal{H})$ with the double dual of ...
- 4,011
2
votes
1
answer
416
views
Fréchet derivative of evaluation-like functional (multivariate)
I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following.
Let $H$ be ...
- 5,632
5
votes
0
answers
264
views
Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?
The spaces $L^2(\mathbb{R})$ (square-integrable functions) and $L^2(\mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $\mathbb{R}$) are two subspaces of the space of ...
- 2,174
2
votes
1
answer
234
views
Approximate selection for finite-valued upper hemicontinuous/semicontinuous maps?
I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite ...
- 60.4k
4
votes
3
answers
242
views
A functional integral inequality
Suppose $f:I=(0,1)\to \mathbb R$ is a continuous function that satisfies
$$ \int_I f(t) e^{at}\,dt \geq 0\quad \text{for all $a \in \mathbb R$}.$$
Does it follow that $f\geq 0$ on $I$?
- 118k
3
votes
1
answer
270
views
Existence of probability measure on the circle with given Fourier coefficients
We say that a Hermitian symmetric (i.e., $f_{-n} = f_n^*$ for any $n \in \mathbb{Z})$ sequence $(f_n)_{n\in \mathbb{Z}}$ is positive-definite if, for any $N \geq 0$ and any $z_0 , \ldots, z_N \in \...
- 60.4k
2
votes
2
answers
211
views
Commutative C*-rings
Let us consider the unital commutative $C^*$-algebra $C[0,1]$. We say $A\subseteq C[0,1]$ forms a C*-subring if it satisfies the following conditions:
1- $A$ is an involutive unital subring (closed ...
- 42.2k
1
vote
0
answers
78
views
Conditions on triangle inequality for integral kernel
Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$.
Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$
A(t,v)=\int_0^{1/v}L(1/t,s)ds,
$$
which is decreasing with $v$ and ...
- 343
2
votes
0
answers
309
views
The conformal map from interior of ellipse to interior of the unit disk (property check)
Based on example 5 (page 546) and example 7 (page 550) of the book "Applied and computational complex analysis. Volume 3, Wiley, 1986" written by Peter Henrici, if $a,b>0$ satisfies $a^2-...
- 700
0
votes
1
answer
185
views
a question about vector valued Banach spaces
I wonder the difference between $L^1(\mu\times\nu)$ and $L^1(\mu;L^1(\nu))$, as if partial derivatives can be exchanged with integration in the second spaces in many articles. In Folland's real ...
- 2,442
16
votes
1
answer
507
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 ...
- 60.4k
3
votes
2
answers
569
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 ...
- 11
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....
- 19.4k
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 $D$ (...
- 18.8k
2
votes
0
answers
89
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 ...
- 217
7
votes
3
answers
618
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 ...
- 42.2k
6
votes
1
answer
616
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 ...
- 61
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 ...
- 739
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
$$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
- 3,390
5
votes
1
answer
149
views
$C^j$-topology considered by Greene and Krantz
My question is about the $C^j$ topology used by Greene and Krantz in their paper "Deformations of Complex Structures, Estimates for the $\bar{\partial}$-equation, and stability of the Bergman ...
- 1,379
10
votes
1
answer
516
views
discontinuous functions on the Sobolev borderline
The Sobolev embedding theorem implies that every function of class $W^{k,p}$ on a reasonable $n$-dimensional domain is continuous if $kp > n$. Cases with $kp=n$ are known as "borderline" ...
- 27.3k
0
votes
0
answers
51
views
Double commutant theorem when $C^*$-subalgebra does not contain identity operator $1$
Double commutant theorem: For a unital $C^*$-subalgebra $M \subset B(H)$, one has
$$\smash{\overline M}^\text{SOT}=\smash{\overline M}^\text{WOT}=M''.$$
My question:
For a $C^*$-subalgebra $M \subset ...
- 11.4k
5
votes
2
answers
220
views
Analytic approximations of smooth vector fields
Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with
$$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$
on $\mathbb{R}^3$ for any $\alpha,K$.
Further, we ...
- 56.6k
2
votes
2
answers
240
views
inequality involving the fractional Sobolev space
Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality
$$|u(x)...
- 407
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 ...
- 15.5k
1
vote
1
answer
314
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 $$\{f_{n}\psi_m(...
- 5,255
3
votes
2
answers
373
views
Norm of two operators on $l^2(\mathbb{Z}_{2}*\mathbb{Z}_{2})$ different
In my research I encounered the following (very concrete) question: Consider the (discrete) group $G:=\mathbb{Z}_{2}*\mathbb{Z}_{2}$. Let $s\text{, }t\in G$ be the generating elements and define for $\...
- 39
2
votes
0
answers
141
views
Uniform bounds for a coupled parabolic system of PDE (linear)
Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon \...
0
votes
0
answers
45
views
Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
- 4,969
1
vote
0
answers
24
views
On some bounds on two constants concerning the disconnectedness of the spectra of small perturbations of operators
Let $H$ be a separable, infinite dimensional, complex Hilbert space. In the book:
Jiang, C. L.; Wang, Z. Y. (1998). Strongly Irreducible Operators on Hilbert
Space. CRC press
above the statement of ...
- 1,165
4
votes
0
answers
133
views
Smooth dependence of solution to elliptic pde depending on parameter
I have a question in mind but let me generalize it slightly.
Suppose I am looking at some pde like
$$-\Delta u + t u = f(u)$$ in $B_1$ (here $u=u(x)$) with $u=0$ on $ \partial B_1$ where $B_1$ is ...
- 1,363
4
votes
1
answer
985
views
Functional derivative of differential entropy
I have trouble finding the derivative of the differential entropy w.r.t the probability density function, i.e. what is $\frac{\delta F[p]}{\delta p(x)}$, where $F[p] = \int_X p(x)\ln(p(x))dx$, and $p(...
- 118k
1
vote
1
answer
410
views
$L^p$ compactness for a sequence of functions from compactness of cut-off
Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
- 2,195
5
votes
1
answer
345
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 ...
- 33.9k
1
vote
1
answer
234
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 ...
- 3,390
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 $\varphi_n:\overline{X}\to X$...
- 5,385
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 ...
- 19.4k
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
$$
\left\|\left[\begin{...
- 3,954
2
votes
0
answers
64
views
Integral convergence with two sequences of functions
I came across this theorem just stated but has not proved and marked by 'it is easy to see'.
Theorem If $u_m$ and $v_m$ converges to $u$ and $v$ in $L^2([0,T];H^1(\Omega))$ weakly and $L^2([0,T];L^2(\...
- 239
2
votes
0
answers
60
views
Uniqueness of solution to Cauchy problem with quadratic nonlinearity
Consider the non-linear differential operator
$$\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\...
- 51
1
vote
1
answer
206
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)=...
- 88.9k
1
vote
0
answers
166
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)$ ...
- 509
5
votes
1
answer
412
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.
...
- 4,744
8
votes
5
answers
483
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!
- 21.8k
1
vote
1
answer
769
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" ...
- 21.8k
3
votes
1
answer
137
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$?
...
- 31.1k
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 ...
- 4,969
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,\...
- 4,969
6
votes
1
answer
218
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?
...
- 9,260
1
vote
3
answers
433
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\}$ ...
- 11.4k