**4**

votes

**0**answers

46 views

### Equivalence of questions regarding restrictions of pure states

In Davidson and Szarek's article "Local Operator Theory, Random Matrices and Banach Spaces" in the Handbook of the Geometry of Banach Spaces, the authors discuss the (now solved) Kadison-Singer ...

**1**

vote

**0**answers

29 views

### Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation

It shall be an old story in PDE.
I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of
$$\inf_{a \in [-1,1]} \{...

**2**

votes

**0**answers

40 views

### How do spectrums interact with bi-Lipschitz maps?

If it makes things simple, we can just stick to bi-Lipschitz maps from $S^k \rightarrow \mathbb{R}^d$ (w.r.t geodesic distance on the sphere with the standard round metric and the $2-$norm on the ...

**0**

votes

**0**answers

77 views

### Dirac functional embedding [on hold]

I got the following set up:
Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with the pointwise ...

**1**

vote

**1**answer

148 views

### Could I affirm that $f$ is not identically 0?

Consider the following situation: Let $\Omega =l^{\infty}(\mathbb{R})$ be the
space of all bounded sequences of real numbers. We will consider in $\Omega$ the metric:
$
d(x,y)=\sum_{i\geq 1}\frac{|...

**2**

votes

**0**answers

80 views

### Measures on a unit sphere of a Hilbert space

Consider a real separable infinite-dimensional Hilbert space $H$. Let $S=\{h\in H\mid \|h\|=1\}$ be a unit sphere in $H$. What are the most natural measures on $S$? Is there a (Borel) measure $\mu$ on ...

**1**

vote

**1**answer

86 views

### Some integrals with respect to a Gaussian measure on a Hilbert space

Assume that $(H,\langle\cdot,\cdot\rangle)$ is a real separable Hilbert space equipped with a Gaussian measure $\mu$ with a mean $m$ and a covariance operator $C$. Let $x\in H$ be a fixed vector. What ...

**4**

votes

**0**answers

58 views

### I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...

**5**

votes

**1**answer

326 views

### Renorming a Banach space to make projections contractive

Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$.
Can the same be done for a family of projections? That is, given finitely many ...

**0**

votes

**0**answers

58 views

### why $\varphi''\in L^{2}(R)$ [closed]

I have the following question: Let $T_{c}$ be an unbounded operator with domain $D(T_{c})=\{u\in L^{2}(R), T_{c}(u)\in L^{2}(R)\}$.
If $\forall \varphi \in \mathcal{C}^{\infty}_{0}(R): \|\varphi''\...

**3**

votes

**1**answer

134 views

### The spectral norm of the truncated exponential of a matrix

Let $A$ be a matrix satisfying $A^*+A\leq0$, it can be shown that $\|e^{tA}\|_2\leq1$ for all $t\geq 0$, where $\|\cdot\|_2$ is the spectral norm defined as largest singular value of the matrix.
I am ...

**1**

vote

**1**answer

101 views

### Gaussian bounds on Dirichlet heat kernel

Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0
$ such that $$\frac{c}{t^{n/2}}
e^{-\frac{1}{4t}d(x, y)^2} \leq ...

**0**

votes

**0**answers

65 views

### Is the Lebesgue measure zero for the discontinuous set of a semicontinuous function? [migrated]

[Q.]
Is there a semicontinuous function, which has its discontinuous set with non-zero measure?
Remark:
Given a semicontinuous function, the set of all discontinuous points may be uncountable, for ...

**-7**

votes

**0**answers

60 views

### W^{∞,p}(IRⁿ) are separable space for 1<p<∞ [closed]

How can prove that the space W^{∞,p}(IRⁿ) are separable space

**0**

votes

**0**answers

72 views

### Looking for an exposition of a certain theorem of Talagrand

The following is a theorem by Talagrand (as stated here, http://arxiv.org/pdf/1511.08609v1.pdf),
Let $(X, \mu)$ be a probability space. Let $F : X \rightarrow \{0,1\}$
be a family of functions ...

**2**

votes

**1**answer

92 views

### $H^1$-continuity of Laplace's equation with respect to boundary data

Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...

**0**

votes

**1**answer

79 views

### Construction of orthonormal basis of the Hilbert space $\mathcal{S}^p_{\mathcal{H}}$ of vectors of $p \in \mathbb{N}$ Hilbert Schmidt operators

Let $(e_j)$ be a orthonormal basis (ONB) of a separable Hilbert space $(\mathcal{H}, \langle\cdot, \cdot\rangle_{\mathcal{H}})$ and $(\mathcal{S_H}, \langle\cdot, \cdot\rangle_{\mathcal{S_H}})$ be the ...

**2**

votes

**1**answer

86 views

### Representation of support of Gaussian measure by kernels of no-variance functionals

Let $\mu$ be a Gaussian measure on a separable Banach space $X$ and $q$ is the covariance operator of $\mu$. I am reading a proof for
$$\operatorname {supp} \mu = \bigcap_{q(f, f) = 0} \ker f =: E$$
...

**0**

votes

**0**answers

24 views

### Convergence of series in inclomplete normed vector space [migrated]

I tried to prove that in non-Banach normed vector space always exists the series that converges absolutely but do not converges.
The idea was to consider Cauchy sequence that don't converges and try ...

**1**

vote

**0**answers

47 views

### Domain of operator

Let be $\lambda\in C^{*}$. Consider the following operator:
$ T_{\lambda}=-\Delta_{R^{2}}++\frac{\lambda^{2} }{4} (x^{2}+y^{2})+i\lambda N$,
where
$N=(x \frac{d }{dy} -y \frac{d }{dx})$ ,
...

**0**

votes

**1**answer

39 views

### characterization of normality by selection theorem

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...

**7**

votes

**3**answers

267 views

### $C^1$-functions on Banach spaces

For Banach spaces $X,Y$ and an open subset $U$ of $X$ a function $f:U\to Y$ is $C^1$ if $U\to L(X,Y)$, $x\to f'(x)$ is continuous where, by definition, the derivative $f'(x)$ is a continuous linear ...

**5**

votes

**0**answers

103 views

### Compactum of Banach algebra

What is an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties?
There exists a bounded approximate identity in $I$ for $I$ i.e., a net $\{e_\alpha\}...

**3**

votes

**2**answers

433 views

### What is a generalized limit?

In the proof of Lemma 1.3 in the paper "The ideal structure of a groupoid C* algebra", Journal of Operator Theory 1991 by Jean Renault, I found the notion of a generalized limit of a net without any ...

**2**

votes

**2**answers

83 views

### Continuous upper envelope of upper semicontinuous function

Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by
$$A = \{\phi \in C(K): \phi \ge u\}.$$
[Q.] Is the following ...

**0**

votes

**1**answer

129 views

### Reproducing Kernel Hilbert Spaces with positive kernels

In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...

**8**

votes

**1**answer

241 views

### Interpolation between $L_1^0$ and $L_2^0$

Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...

**-1**

votes

**0**answers

12 views

### Continuous function on compact topological space [migrated]

I came across the following statement.
Let $X$ a uncountable set, $p \notin X$ and $X^* = X \cup \{p\}$. Let $$\mathcal O := \{O \subseteq X^* \mid O \subseteq X \text{ or } p \in O \text{ and } X \...

**2**

votes

**1**answer

49 views

### Transformation inverting distances between two sets of diameter 1

Let $S_1, S_2 \subseteq \mathbb{R}^2$ be two finite disjoint sets of points in the plane with $\texttt{diam}(S_1) \leq 1$ and $\texttt{diam}(S_1) \leq 1$.
Does there always exist a transformation $f: ...

**3**

votes

**2**answers

111 views

### Compact embeddings between vector-valued Holder spaces

Let $S\subset\mathbb{R}^n$ be compact, $\alpha,\beta\in(0,1)$, $\alpha>\beta$ and $X$ a Banach space.
Under which assumptions on $X$ is the embedding
$$C^\alpha(S;X)\subset C^\beta(S;X)$$ compact?
...

**5**

votes

**1**answer

190 views

### finding subharmonic function on the ball with both Dirichlet and Neumann boundaries prescribed

I have a question which looks like some sort of inverse problem.
Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$).
Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) ...

**-1**

votes

**0**answers

28 views

### Problem regarding continuous embeddings [migrated]

Given the following exercise:
We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \{x \in (\mathbb R_+) : f(x) \neq 0\} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$
and for all $f \in C^...

**3**

votes

**0**answers

97 views

### quasi-nilpotent part of a dual operator

Definitions and notation.
Let $X$ be a complex Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator on $X$. We define the quasi-nilpotent part of $T$ as
\begin{equation*}H_0(T):=\left\{...

**2**

votes

**1**answer

172 views

### A unital algebra with norm and continuous multiplication is a Banach algebra

In my research in functional analysis, I came across this rather simple result:
For a normed algebra A over $ \mathbb{C} $ with unit, in which multiplication , right and left are both continuous w....

**0**

votes

**0**answers

61 views

### On the transitivity of the action of the unitary group

Let $H$ be a complete inner product space over either real or complex numbers. If $H$ is complete, for two finite sets of vectors $\left\{e_i\right\}_{i\in I}$ and $\left\{f_i\right\}_{i\in I}$ there ...

**5**

votes

**1**answer

153 views

### Convergence of functionals on compact projections on a separable Hilbert space

Let $H$ be a separable Hilbert space over $\mathbb{C}$, say $\ell_2$ for simplicity. Let $\mathcal{K}(H)$ denote the space of all compact operators on $H$ and $\mathcal{P}(H)$ the set of all finite ...

**5**

votes

**3**answers

160 views

### Morrey's inequality for Sobolev spaces of fractional order

Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions (or distributions), $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that
$$
\|u\|_{H^s}^2=\sum_{k\...

**5**

votes

**0**answers

80 views

### Generalized singular numbers and the Haagerup $L^p$ spaces

Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...

**0**

votes

**0**answers

75 views

### Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$

Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...

**0**

votes

**0**answers

108 views

### A topology on the product space of Euclidean space and smooth functions space

I'd like to know if there is a well-known topology on the space $S := \mathbb R \times C^\infty(\mathbb R)$, such that $(x_n, f_n) \to (x, f)$ in $S$ with respect the topology is equivalent to
$$(x_n,...

**3**

votes

**1**answer

88 views

### almost invariant half space for a dual of a restricted operator

Let $X$ be an infinite-dimensional Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. A closed subspace $Y$ of $X$ is said to be an almost-invariant halfspace (...

**19**

votes

**1**answer

435 views

### What is the Banach-Mazur distance between $\ell_\infty$ and $L_\infty$?

Given Banach spaces $X$ and $Y$, the Banach-Mazur distance between $X$ and $Y$ is defined as
$$ d(X,Y) = \inf\{ \|\varphi\|\|\varphi^{-1}\| : \varphi\colon X\to Y \text{ isomorphism} \}.
$$
We ...

**5**

votes

**0**answers

77 views

### Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...

**1**

vote

**0**answers

37 views

### Minimization over function space [closed]

This is the first time I encounter a problem where I need to minimize a function defined in a function space. Define the function space $A=\{\theta:\mathbb{R}^3 \rightarrow\mathbb{R}|\ \theta\ \text{...

**10**

votes

**1**answer

253 views

### Sequence of nested sets in $[0, 1]$ with bound on gaps

What is the best possible $\epsilon$ and sequence $(a_n)_{n = 1}^\infty \subset [0, 1]$ we can find such that
$$
d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in ...

**0**

votes

**1**answer

87 views

### Some questions related to the unitary operators

A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product.
What is the name of the analogue for the real case? Orthogonal operator ...

**1**

vote

**1**answer

50 views

### Name for a uniform local boundedness property of a function

I am working with a function $f : \mathbb{R}^N \to \mathbb{R}$ having the property that for every $R > 0$, there exists $M > 0$ such that if $x, y \in \mathbb{R}^N$ and $\vert x - y \vert \le R$,...

**1**

vote

**1**answer

142 views

### Lipschitz functions and $W^{1,\infty}$

I am not sure my question is research type, but I am sure I can find here an answer.
So we have the following theorem in the book of Lawrence Evans in PDE, 2nd edition pages 294-295:
Theorem 4 (...

**5**

votes

**0**answers

151 views

### Extension operators for topological vector space-valued smooth functions on closed sets

There are many known results about extension theorems for real-valued functions on closed sets, with varying levels of differentiability and so on, all very roughly following the Whitney approach. For ...