All Questions
10,448 questions
0
votes
1
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208
views
Can a hyperplane be contained in a subspace?
Suppose $Y$ is a subspace of a normed linear space $X$ and let $y\in S_Y, y^*\in S_{Y^*}$ such that $y^*(y)=1$, where $S_Y$ denotes the closed unit sphere in $Y$. My question is the following:
Is it ...
- 585
0
votes
1
answer
853
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)...
- 698
0
votes
1
answer
288
views
Boundness and adjoint of an integration operator
Let's define the operator $K$ as
$$
\begin{eqnarray*}
K :&H&\longrightarrow &\;\:Z&\\
&y\left( x;\sigma \right) &\mapsto &\!\!\!\!\int_{0}^{1}C\left( \sigma \right)
y\left(...
0
votes
1
answer
80
views
Vectors concentrated on one coordinate
Suppose $X$ is a Banach space, $(e_i)$ a normalized basis, $(e_i^*)$ the biorthogonal functionals, and $Y$ a finite codimensional subspace of $X$. Given $N$ and $\varepsilon$, can we find $x\in Y$ ...
- 247
0
votes
1
answer
116
views
Integrable function [closed]
Suppose that $a, b, c_1$ and $c_2$ are real constant.
Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e.
$$\int_1^{\infty}\int_1^{\...
- 119
0
votes
1
answer
314
views
Regarding orthogonality in Banach space
Let $(X,\|.\|)$ be a Banach space over the real line. Let $x\neq 0$ and $y\neq 0$ be in $X$, then $x$ is said to be orthogonal to $y$ if $\|x+\lambda y\|\geq\|x\|$ for every real number $\lambda$.
...
- 195
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
views
$f_n$ 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
votes
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
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
views
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 $n$-...
- 7,789
0
votes
1
answer
87
views
On an error bound for matrix constraints
Given $A,B\in\mathbb R^{n\times n}$ such that there is an $U\in\mathbb R^{n\times n}$ with $UU'=I$ and $UAU'=B$.
Suppose each entry of $V$ is within $\pm\epsilon$ of each entry of $U$ is there a way ...
- 13.9k
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
0
votes
1
answer
198
views
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
0
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
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
0
votes
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
61
views
Define a matrix function with a specific property
Let $S$ be the set of all positive semidefinite Hermitian matrices of order $mn$ over $\mathbb{C}$. Any matrix $H$ can be partitioned into blocks $H_{ij}$ of order $n$ that is $H_{mn \times mn} = (H_{...
- 363
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
votes
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$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1,...
user81900
0
votes
1
answer
693
views
the double dual of "little l one" sequence space
I remember a professor remarking a while back that the double dual of the sequence space $l_1^{\infty}(\mathbb{R})$ is a very complicated space. I understand it must contain a copy of the original ...
- 9
0
votes
1
answer
326
views
On the Riesz representation theorem II
I have a follow-up question to On the Riesz representation theorem .
Let $V$ be a subspace of a Hilbert space, and let $V^\times$ be the space of all antilinear functionals on $V$, equipped with the ...
- 3,873
0
votes
1
answer
770
views
About weak derivatives [closed]
I have a question about weak derivatives.
Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some
open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...
- 721
0
votes
1
answer
203
views
Continuity of Functional Represented by Surface Integral
Let $\Omega \subset \mathbb{R}^n$ be open and bounded and let $S \subset \Omega$ be a hypersurface in $\mathbb{R}^n$. Let further be $C_0(\Omega)$ the space of all continuous functions with compact ...
- 53
0
votes
1
answer
559
views
Densely-defined unbounded operators with large support
Most densely-defined unbounded linear operators on Hilbert spaces have a very large domain. In fact, for a lot of natural operators the intersection of their domains are still dense.
Let us consider ...
- 5,327
0
votes
1
answer
254
views
Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases}
-\...
- 266
0
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1
answer
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
2
answers
737
views
Eigenvalues of an amplification matrix
Let $A$ and $B$ square real matrices.
I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1.
Can we say something about the eigenvalues ...
- 1
0
votes
1
answer
276
views
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
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
votes
1
answer
454
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
0
votes
1
answer
222
views
Bounding near the boundary for a Sobolev function.
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>\frac{d}{2}$ (i.e. $f$...
- 3,217
0
votes
1
answer
126
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,153
0
votes
1
answer
119
views
Nonstationary phase method for oscillatory integral
I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth.
The stationary phase method says that if $t_0\in [a,b]$ is such that ...
- 107
0
votes
1
answer
188
views
Does the second Bourgain–Delbaen space belong to C_p?
The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$.
An operator $T:X\to Y$ ...
