**1**

vote

**0**answers

52 views

### Has every Lusin vector space a stronger Polish vector space topology?

Let $X$ be a topological vector space or even a locally convex space such that its (vector space) topology is Lusin, i.e. there is some stronger Polish topology. Does there also exist a stronger ...

**0**

votes

**1**answer

52 views

### On a theorem by Mooney and Khavin on the weak sequential completeness of the predual of $H^\infty(\mathbb{D})$

There is a theorem by Mooney http://msp.org/pjm/1972/43-2/pjm-v43-n2-p.pdf#page=185 and independently proved by Havin which says that the predual of $H^{\infty}(\mathbb{D}),$
$L^{1}/H^{1}_{0}$ is ...

**3**

votes

**1**answer

102 views

### A clarification regarding analytic perturbation of metrics and Laplacian

This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...

**2**

votes

**0**answers

58 views

### Lower semi-continuity of the Hellinger-Fisher-Rao distance

I am currently working on unbalanced optimal transport, where the Hellinger (or sometimes Fisher-Rao) distance
$$
...

**0**

votes

**0**answers

91 views

### from finite to $\sigma$-finite measure space [migrated]

This might be rather elementary. I have put it at MSE for a while without getting any answers.
Here is the question:
In the proof of the following theorem, would anyone explain how the general case ...

**11**

votes

**2**answers

282 views

### When is the closed unit ball in a smaller Banach space closed in a larger Banach space?

Recently I saw an interesting lemma:
For any $s>0$, the closed unit ball in $H^s$ is also closed in the $L^2$ norm. That is, suppose $u_j\in H^s$ and $\|u_j\|_{H^s}\le 1$. Suppose $u_j\to u$ in ...

**1**

vote

**0**answers

57 views

### Asymptotics of a elliptic pde when exponent gets large

I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball ...

**-8**

votes

**0**answers

50 views

### connectedness in Euclidean Space [closed]

f:[0,1] to R is a map defined by f(x)=sin(1/x),for x is not zero and
f(0)=0.X={(a,f(a))|a is in [0,1]}.prove that,X is a connected subset of
R^2 where the metric is usual euclidean metric on R^2.

**-6**

votes

**0**answers

66 views

### Continuity of Real line [closed]

f:R to R such that f attains every value exactly twice i.e. for all a in
R, {x in R|f(x)=a} is either empty or doubleton set.prove that,f is
discontinuous at infinitely many points .

**5**

votes

**2**answers

220 views

### Thin large subspaces of $\ell^N_1$

Consider a sequence $V_N$ of subspaces of $\ell^N_1$ so that $\dim V_N = N- n$ and $n$ is $\mathsf{o}(N)$. Is it true that these spaces are "thick" (unofficial terminology), i.e. are there constants ...

**2**

votes

**0**answers

74 views

### A modification of Minty's trick?

I have the following result:
$$0 \leq \int_0^T (a(t)- |w(t)|)(b(t) - g^{-1}(|w(t)|))\quad\forall w \in L^2(0,T)$$
where $a$ and $b$ are both non-negative.
Does it follow that $b(t) = g^{-1}(a(t))$? ...

**0**

votes

**0**answers

103 views

### What is the name of this theta related function? [closed]

My adviser showed me a function which is important for my current work, but he doesn't remember where he got this. He just knows it's a kind of theta function. I cannot find any literature talking ...

**6**

votes

**2**answers

225 views

### Completeness of nonharmonic Fourier Series

I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system ...

**2**

votes

**0**answers

77 views

### Heat equation - regularity of solutions [migrated]

Consider the heat equation on $\mathbb{R}$
$$
u_t=u_{xx}
$$
with boundary conditions $u(0,x)=g(x)$.
It is well-known that even if the function $g$ is "very bad'' (say, only bounded but not ...

**-3**

votes

**0**answers

48 views

### What is operator valued function? [closed]

I'm new to functional analysis and confused what operator valued means.

**-1**

votes

**0**answers

58 views

### Product of smooth functions [closed]

Let $O$ be a non empty open subset of a bounded open set $\Omega\subset R^n,$ and let $f\in L^2(\Omega)\cap W^{2,\infty}(O)$.
Let $u: \Omega \to R$ be a function such that the function $g$ defined ...

**4**

votes

**0**answers

68 views

### Smooth perturbation of a positive self-adjoint operator with compact resolvent

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...

**0**

votes

**0**answers

44 views

### Left introversion operators associated to function spaces on semigroups

I am stuck on the following question for quite sometime now. Please help, any comment is welcome.
Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...

**3**

votes

**2**answers

143 views

### Lower bound for Euler's function

Euler function is defined, for $|x|\le 1$, as follows:
$$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$
Upper bounds for $\phi$ can be simply derived from ending the product early, e.g.
...

**3**

votes

**0**answers

293 views

+50

### Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention.
Note 2: This is my research problem, but the original problem ...

**0**

votes

**0**answers

29 views

### How the diffusion in the unit ball induce the boundary process on the boundary directly?

This is Example 1.2.3 from Fukushima, Masatoshi, Oshima, Yoichi and Takeda, Masayoshi's book "Dirichlet Forms and Symmetric Markov Processes".
In this example, we define a Dirichlet form in the unit ...

**3**

votes

**0**answers

120 views

### Proof without distributions

I was wondering whether there is a way to show this identity
$$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$ without using distributions for $f ...

**5**

votes

**1**answer

97 views

### Is the Feichtinger's algebra $(S_0(\mathbb{R^d}),||\cdot||_{S_0})$ reflexive?

The Feichtinger's algebra
$S_0(\mathbb{R^d})=M^{1,1}(\mathbb{R^d}):=\{f\in L^2(\mathbb{R^d}):V_g(f)\in L^1(\mathbb{R^{2d}})\}$, where
$V_g(f)(x,\omega)$ is the short-time Fourier transform of $f$ with ...

**9**

votes

**1**answer

193 views

### The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem

Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, ...

**6**

votes

**0**answers

143 views

### Commutation preserving operators

Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve ...

**3**

votes

**0**answers

90 views

### Eigenvalues and eigenvectors of the q-Bernstein operator

The Bernstein operator maps $f\in C[0,1]$ to its Bernstein
polynomial $B_n f.$ The eigenvalues and eigenfunctions of the
Bernstein operator on $C[0,1]$ have been described in [1]. Similar description ...

**4**

votes

**1**answer

112 views

### Invariant subspaces are reducing subspaces in $L^2(\mu)$; where $\mu$ is a singular measure w.r.t Lebesgue measure

I have already posted this question on math.stackexchange but didn't get any answer. I hope this is the right place to ask this question.
Recently I was reading a book "Operator Function and system" ...

**1**

vote

**1**answer

106 views

### Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra

Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of ...

**1**

vote

**0**answers

66 views

### One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...

**0**

votes

**1**answer

111 views

### Nonstable $K$-theory question

Let $Y$ be a compact, Hausdorff topological space, and $X$ be a locally compact, contractible, Hausdorff space which is homeomorphic to a dense subset of $Y$.
Question A: Is ...

**1**

vote

**0**answers

41 views

### Operator algebra generalization of linear algebra result on diagonalization of commuting operators with distinct eigenvalues [migrated]

In linear algebra it is true that: a) if $\mathcal{A}$ is a set of unitarily diagonalizable matrices (in $\mathbb{C}$, i.e. normal matrices) that commute with each other then they are simultaneously ...

**4**

votes

**1**answer

118 views

### Is Hessian operator self-adjoint on infinite dimensional environment?

As we know the Hessian matrix is symmetric in a finite-dimensional environment. What about the Hessian operator $D^2F$ for a functional $F:H\rightarrow \mathbb{R}$, where $H$ is a Hilbert space and ...

**0**

votes

**1**answer

112 views

### Averaging measurable functions over amenable group actions

Let $G$ be an amenable group acting on a space $X$.
Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$.
Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...

**3**

votes

**0**answers

109 views

**0**

votes

**1**answer

82 views

### The monotone operator in $BV$ space

I am considering the following minimizing problem:
$$
\min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\}
$$
where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...

**1**

vote

**0**answers

34 views

### Mathematical difference between broad and narrow band Spectral estimation [closed]

Is there different mathematical formulation behind spectral estimation of narrow band and wide band? By spectral estimation I mean estimating the frequencies in a given signal. Fourier transform is ...

**3**

votes

**0**answers

93 views

### Differential operators acting on the Schwartz space

I asked a similar question on math stack exchange but didn't get an answer so I will try to ask it here. Any help/suggestion is most than welcome!
Let $D$ be a linear differential operator with ...

**2**

votes

**0**answers

28 views

### Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$

Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...

**0**

votes

**0**answers

42 views

### The properties of the solution pf minimizing problem with different parameters

I asked an similar problem before but received no respond. Here I modified the problem, add in more informations and assumptions, and with an extra question...
Let $\Omega\subset \mathbb R^2$ be open ...

**1**

vote

**1**answer

185 views

### How are the assumptions used in the proof of Bourbaki-Alaoglu Theorem?

I've posted this question in MSE for a while without getting any answer.
In the proof of the following theorem, where are the assumptions "Hausdoff" and "locally convex" used?

**2**

votes

**0**answers

115 views

### Is $C^{\gamma}(\Sigma)$ dense in $C(\Sigma)$?

Consider $\{0,1\}$ with the discrete topology and $\Sigma=\{0,1\}^{\mathbb{N}}$ with the product topology. We know that this
product topology is generated by the metric
$$
...

**0**

votes

**0**answers

125 views

### Amenable hypergroups

I needed some information about the intersection of the kernels of invariant means on hypergroups. So I read the discussion made for the question " The kernel of all invariant means " which answer my ...

**3**

votes

**3**answers

132 views

### Characterizing (up to permutations) finite sequences of real numbers

Let $S=\{x_1,\cdots, x_N\}$ be a finite sequence of real numbers.
I am interested in characterizing the family of functions $F$ such that for any $f\in F$ the function
$$
c(\lambda) ...

**1**

vote

**1**answer

98 views

### Comparison between spectra

Let $G$ be a normal operator with compact resolvent on a Hilbert space $H$ such that ${\rm ker}(G) \neq {0}$. Further let $P$ be the orthogonal projection onto ${\rm ker}(G)$, and let $G_{0}:=G+P$.
...

**5**

votes

**0**answers

167 views

### Constructive approximation of Lipschitz functions

There are a number of theorems in classical functional analysis about approximation of Lipschitz functions by smooth functions. I was wondering if there are any similar constructive and explicit ...

**3**

votes

**1**answer

186 views

### On the embedding of a function space $X$ into $L^2\cap L^4$

It is well-known that if $\Omega\in \mathbb{R}^n$ is a bounded domain, then we have the embedding
$$
L^4({\Omega})\subset L^2({\Omega})
$$
since $||f||_{L^2(\Omega)}\leq C(\Omega) ||f||_{L^4(\Omega)}$ ...

**16**

votes

**2**answers

372 views

### Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and ...

**8**

votes

**2**answers

281 views

### Constructing a function over a metric space through given points

Suppose there is a compact metric space $(X,\rho)$ and a Euclidean space $\mathbb{R}^n$.
There is a sequence of unequal points $\{x_1,...x_N\}$ in $X$ such that all metrics $\rho(x_i,x_j)$ are known ...

**1**

vote

**0**answers

42 views

### Existence and uniqueness of Abel integral equation

I consider the following Abel's integral equation:
$$
\int_0^t \frac{k(t,s)f(s)}{\sqrt{t-s}}=g(s)
$$
where $g(s)\in C^{\infty}[0,T]$ and $k(t,s)=C+\sqrt{t-s}$.
To the best of my knowledge, there ...

**8**

votes

**0**answers

113 views

### Optimal exponent in the Lojasiewicz-Simon gradient inequality

Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, ...