All Questions
13,927 questions
0
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208
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strict topology on multiplier algebras
Suppose $A$ is a $C^*$ algebra,$M(A)$ is the multiplier algebra.If $S$ is a subset of $M(A)$ which is compact for the strict topology on $M(A)$,is $S$ also a subset of $M(M(A))$ which is compact for ...
- 451
0
votes
1
answer
119
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$\sup_{f} \inf_{z\in D} [f_x^2(z)+f_y^2(z)]$ for $|f|\leq1$ on a unit disk
Let $f:\mathbb{R^2}\mapsto\mathbb{R}$ be continuous and have partial derivatives in $D=\{(x,y):x^2+y^2\leq1\}$, and let $\mathscr{H}$ the set of such functions for which $\sup_D |f|\leq1$.
Could ...
- 385
0
votes
1
answer
87
views
Convergence of self-adjoint elements in $\sigma$-weak topology
In a von Neumann algebra, if $A_{\alpha}$ converges to $0$ in the $\sigma$-weak topology, do the positive parts $(A_{\alpha})_{+}$ necessarily converge to $0$ in the $\sigma$-weak topology?
- 159
0
votes
1
answer
131
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$f_n$ is bounded in $C(0,T;H^2(0,L))$ so is $f_n^p$?
Let $1<p<\infty$, and $f_n$ be a bounded sequence in $C(0,T;H^2(0,L))$. It looks obvious to me that $f_n^p$ is also bounded in $C(0,T;H^2(0,L))$. When we take the derivative of $f^p(t)$ twice we ...
- 395
0
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1
answer
66
views
Covergent net in $\mathcal{E}'(\mathbb{R})$ implies bounded?
Let $\mathcal{E}(\mathbb{R})$ be the space of all $C^\infty$ functions on $\mathbb{R}$ with its usual topology, and $\mathcal{E}'(\mathbb{R})$ be the dual space with the weak* topology.
Let $(T_i)_{...
- 11
0
votes
1
answer
274
views
Continuity of maps in which preimage preserves compactness
Let $X$ and $Y$ be Hausdorff spaces and suppose that $Y$ is locally compact. Let $f:X\to Y$ be a surjective map such that for any compact subset $K \subset Y$ the pre-image $$f^{-1}(K)=\{x\in X: f(x)\...
- 563
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2
answers
1k
views
Does point-wise weak convergence give weak convergence in $L^2(I;X)$?
Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...
- 395
0
votes
1
answer
81
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Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$?
Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:
$$u_t = grad[V(u)]$$
For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-...
- 7,799
0
votes
1
answer
282
views
Does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and …?
Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with non-zero finite length $L$. Then, does there always exist a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
- 113
0
votes
2
answers
494
views
Semifinite measure and spectral theorem
Let $H$ be a complex Hilbert space (not necessary separable).
Spectral Theorem: Let $A_1$ and $A_2$ be two commuting normal operators, then there exists a measure space $(X,\mathcal{E},\mu)$,
two ...
- 1,154
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1
answer
198
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Doubt in proof of $\lim_{n \to \infty} [\lambda R(\lambda, A)]^n = P.$
I have a doubt in proof of Lemma $4.7$ of this paper.
Lemma: Let $A$ be a closed operator on a complex Banach space $E$ and assume that $0$ is an eigenvalue of $A$ and a pole of the resolvent $R(\...
- 343
0
votes
1
answer
188
views
Projective tensor product
Let $A$ and $B$ be Banach algebras. Then the map $\phi:(A\widehat\otimes A) \oplus_\infty (B\widehat\otimes B) \to (A\oplus_\infty B)\widehat\otimes(A\oplus_\infty B)$ is a contractive embedding.
Can ...
- 565
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votes
3
answers
554
views
Converting a bounded metric into an unbounded metric
Suppose $d$ is a bounded metric on $X$, i.e. $d(x,y)< K<\infty$ for all $x,y\in X$. Is there a standard way to convert $d$ into another metric $\widetilde{d}$ on $X$ with the property that $\...
- 710
0
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1
answer
129
views
Is there a $\sigma$-metacompact space which is not metacompact?
Recall that a space $X$ is metaLindelof if every open cover of
$X$ has a point-countable open refinement.
A space $X$ is metacompact if every open cover of
$X$ has a point-finite open refinement....
- 621
0
votes
1
answer
268
views
Spectral Theorem, $AB = BA \implies B\Phi(f) = \Phi(f)B$
I'm studying the spectral theorem as appears in Reed and Simon's Functional Analysis.
Assume we have constructed the continuous functional calculus for a self adjoint bounded operator $A$ on a ...
- 181
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1
answer
165
views
Is there an example of a one to one and onto mapping between these two spaces?
Let $\Omega$ be a convex open subset of $\mathbb{R}^d$ with a smooth boundary. Is there an example of a one to one and onto mapping of the form $$L^{d+1}(\Omega) \to W^{1,d+1}(\Omega)$$
- 698
0
votes
1
answer
103
views
Continuous orthogonal preserving maps between projective space
Is there a continuous map $f:\mathbb{C}P^n \to \mathbb{C}P^m$, for some $n>m$
which preserve orthogonality?Namely $x\perp y \implies f(x) \perp f(y) $?
If yes, are there two non homotopic ...
- 366
0
votes
1
answer
668
views
A possible proof of the Borsuk Ulam theorem without "Homology-Cohomology"
Assume that $n>1$.
The configuration space of $S^n$ is defined as follows $$M_n=\{(x,y)\in S^n\times S^n\mid x \neq y\}$$
We have two questions:
1.Is there a continuous function $f:M_n ...
- 366
0
votes
1
answer
57
views
Refining ultraconnected spaces to connected $T_2$ spaces
Let $X$ be an infinite set and suppose $\tau$ is an ultraconnected topology on $X$ without isolated points. Is there a topology $\sigma\supseteq \tau$ such that $(X,\sigma)$ is a connected $T_2$-space?...
- 48.9k
0
votes
1
answer
303
views
Approximation of a $C^{\infty}_c$ function with tensor products of a constant tensor rank
I asked the following question a few days ago:
Approximation of a $C^{\infty}_c$ function by tensor products
However, I then realised that I actually need a stronger result in my proof.
As in the ...
- 357
0
votes
1
answer
388
views
Why do the eigenfunctions of a 1D Schroedinger operator with even potential alternate in parity?
Let $\mathcal L$ be a Schroedinger operator on the real line of the form
$\mathcal L = -\frac {d^2} {dx^2} + V(x),$
where $V$ is an even, smooth function. I am interested in the case where $V(x)\to ...
user101142
0
votes
2
answers
366
views
About density of some subsets of infinitely differentiable functions in $C[0,1]$
Let $x_1,...,x_m$ be fixed numbers from $[0,1]$ and let $k_1,..., k_m$ be fixed natural numbers ($\geq 1$).
Is the set
$$\{f\in C^\infty[0,1]: f^{(k_1)}(x_1)=0,...,f^{(k_m)}(x_m)=0 \}$$a dense subset ...
- 3
0
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1
answer
629
views
Fourier Transform of sub-Gaussian distributions
The high level question is: Just as the Fourier transform of a Gaussian is a Gaussian, is the Fourier Transform of a sub-Gaussian also a sub-Gaussian?
Let $x \in \mathbf{R}^n$ denote some sub-...
- 445
0
votes
1
answer
138
views
Extracting a subsequence for which $\sup_j \left\vert \text{supp}(x_{n_j}) \right\vert < \infty$
Let $X$ be a Banach space with basis $(e_n)_{n=1}^\infty$, and suppose that $(x_i)_{i=1}^\infty$ is a normalized block basic sequence of $(e_n)_{n=1}^\infty$. In addition assume that $(x_i)_{i=1}^\...
- 125
0
votes
3
answers
320
views
Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? [closed]
Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1,...
user81900
0
votes
1
answer
697
views
the double dual of "little l one" sequence space
I remember a professor remarking a while back that the double dual of the sequence space $l_1^{\infty}(\mathbb{R})$ is a very complicated space. I understand it must contain a copy of the original ...
- 9
0
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1
answer
326
views
On the Riesz representation theorem II
I have a follow-up question to On the Riesz representation theorem .
Let $V$ be a subspace of a Hilbert space, and let $V^\times$ be the space of all antilinear functionals on $V$, equipped with the ...
- 3,873
0
votes
1
answer
74
views
$T_2$-spaces such that the lattices of open sets can be embedded into each other
Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$.
Does this imply that $(X,\tau)\cong (Y,\sigma)$?
user62017
0
votes
1
answer
770
views
About weak derivatives [closed]
I have a question about weak derivatives.
Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some
open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...
- 721
0
votes
1
answer
203
views
Continuity of Functional Represented by Surface Integral
Let $\Omega \subset \mathbb{R}^n$ be open and bounded and let $S \subset \Omega$ be a hypersurface in $\mathbb{R}^n$. Let further be $C_0(\Omega)$ the space of all continuous functions with compact ...
- 53
0
votes
1
answer
559
views
Densely-defined unbounded operators with large support
Most densely-defined unbounded linear operators on Hilbert spaces have a very large domain. In fact, for a lot of natural operators the intersection of their domains are still dense.
Let us consider ...
- 5,327
0
votes
1
answer
254
views
Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases}
-\...
- 266
0
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1
answer
384
views
Heisenberg group acts on the circle
Let $H$ be a Heisenberg group, i.e.
$$
H=\left\langle a,b,c |[a,b]=c,[a,c]=[b,c]=1\right\rangle.
$$
$H$ acts on the circle by homeomorphism which preserves the orientation. If the rotation number of $...
- 21
0
votes
1
answer
272
views
Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?
Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space,
$$A(\mathbb T):= \{f\in L^...
- 1,051
0
votes
2
answers
225
views
Isomorphism theorem for subfactors?
It's about the existence of a generalization of the first isomorphism theorem for groups, for subfactors :
Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II_1$ ...
0
votes
1
answer
134
views
Two different products of filters
By filters I will mean filters on some set $\mho$.
I define product of an infinite family of filters in two ways. I feel (by analogy with properties of Tychonoff product vs box product of topological ...
- 765
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1
answer
237
views
How to see such space is Lindelof?
Let $R$ denote the set of all real numbers. $B$ is any Bernstein set of $R$.
Bernstein Set: A subset of the real line that meets every uncountable closed subset of the real line but that contains ...
- 654
0
votes
3
answers
1k
views
Zero-dimensional space
Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A_{\alpha} : \alpha\in S\}$ such that $\overline{A}=\overline{\...
- 192
0
votes
1
answer
276
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Positive definite Hermitian matrices of countable rank
Say that a $\omega\times \omega$ Hermitian matrix $A$ is positive semidefinite of rank $n$ if there exists a $\omega\times n$ complex matrix $B$ such that $A=B B^\dagger$ where $^\dagger$ denotes the ...
user2529
0
votes
1
answer
455
views
Sequences satisfying $x_{n+1} \geq \alpha (x_1+\ldots{} +x_{n})$ for $\alpha<1$
Consider a positive sequence $x_n >0$ that satisfy the condition that there exists a constant $0<\alpha<1$ such that $x_{n+1} \geq \alpha (x_1+\ldots{} +x_{n})$.
What can be said about the ...
- 81
0
votes
1
answer
382
views
Double duals characteristic [closed]
Recall that (for $1\le p<\infty$), $\ell^p = \{\{a_n\}_{n=1}^\infty:\sum\limits_{i=1}^\infty|a_i|\lt\infty\}$, with norm $||\{a_n\}||=(\sum\limits_{i=1}^\infty|a_i|^p)^{\frac{1}{p}}$.
It is well ...
- 13
0
votes
1
answer
193
views
Dissipative operator
Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative.
is it true that : if $\|y\|\leq \|z\|$ then $\|Ay\|\leq \|Az\|$?
Thank ...
- 11
0
votes
1
answer
338
views
Ultraweak closure inside a closed ball
Let $H$ be a Hilbert space, and $S\subseteq \mathcal{B}(H)$. We denote
$\bar S$ the ultraweak closure of $S$, and $B_r$ the closed ball of center 0
and radius $r>0$ of the normed space $\mathcal{...
- 33
0
votes
1
answer
612
views
Calculating a distributional derivative
Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
- 213
0
votes
2
answers
1k
views
Weak versus strong convergence
This is my first time posting.
I am well aware that an $L^2$ weakly converging sequence is not convergent in the corresponding strong topology. However, my question is as follows, do the sequence of ...
- 213
0
votes
1
answer
278
views
On the compactness of a certain chain topology [closed]
Let $X$ be a non-empty set and $I$ a collection of some nested subsets of $X$ indexed by a linearly ordered set $(\Lambda,\le)$ such that $I$ always contains the void set $\emptyset$ and the whole set ...
- 243
0
votes
1
answer
286
views
Irreducible subspaces of separable Hilbert space
A question about definition: Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, with $B(\mathcal{H})$ the bounded linear operators on it. What does it mean to have an irreducible ...
- 103
0
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1
answer
455
views
Is this set of functions compact?
Let $\mathcal{F}$ be the set of continuous functions $\varphi$ from $\mathbb{C}$ to $[0,1]$ that satisfy $\begin{align}\varphi(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\varphi(z+e^{i\theta})d\theta\end{align}$ ...
- 1
0
votes
1
answer
622
views
products in the category of banach spaces
Let $\{X_{\alpha} \}_{\alpha \in A}$ be a collection of Banach spaces. It is easy to show that $ P = \{(x_{\alpha}) : {\rm sup}_{\alpha} \|x_{\alpha} \| < \infty \} $ with $\| (x_{\alpha} ) \| = {\...
- 3,830
0
votes
1
answer
2k
views
separability of a certain space of continuous functions
Let $O$ be an open subset of the separable Hilbert space $H.$ Let $E$ be a separable Banach space. Is it true that $C^0_b(O;E),$ the space of bounded continuous maps $O\rightarrow E$, endowed with ...
- 2,935