# Tagged Questions

**7**

votes

**3**answers

302 views

### Rosenthal like inequality for weak $\mathbb L^p$-norms

Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if ...

**4**

votes

**1**answer

224 views

### Approximation of an integral over the unit ball of L_1

For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and
$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- ...

**1**

vote

**1**answer

199 views

### Agreement of two topologies on a linear space

I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.
Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...

**4**

votes

**3**answers

544 views

### Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions?

Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...

**5**

votes

**0**answers

152 views

### Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...

**4**

votes

**1**answer

195 views

### Gaussian Valued Random Variables in Geometry of Banach Spaces

Why are Gaussian valued random variables so important in the Geometry of Banach spaces? I am reading the monograph by Pisier - "Probabilistic Methods in the Geometry of Banach Spaces" and in the very ...

**4**

votes

**1**answer

153 views

### Set of unitaries with “spread-like” properties

I'm interested in finding two sets of $N$ unitary $N \times N$ matrices $U_{1}, \ldots, U_{N}$, $V_{1}, \ldots, V_{N}$ such that:
$
\sup\limits_{X, Y}\sum\limits_{j,k = 1}^{N} ...

**5**

votes

**1**answer

383 views

### Derandomizing random matrices

My question is rather general - what is known about derandomization of results in random matrix theory, high-dimensional geometry, Banach spaces etc. using probabilistic constructions (like estimates ...

**6**

votes

**1**answer

272 views

### Best constant in comparison between Rademacher and gaussian averages?

Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables.
What is the best ...

**1**

vote

**1**answer

520 views

### If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$?

Let $H$ be an infinite-dimensional, separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Let $H^*$ denote the space ...

**6**

votes

**1**answer

721 views

### If $H$ is a separable Hilbert space, is $L^2(H)$ separable?

Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?

**6**

votes

**1**answer

363 views

### The typical size of a random element in a Banach space

Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...

**3**

votes

**1**answer

712 views

### Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space

This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...

**16**

votes

**5**answers

2k views

### Brownian motion and spheres

Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$:
$$ ...

**3**

votes

**2**answers

748 views

### Convergence of Gaussian measures

Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian ...

**7**

votes

**3**answers

711 views

### Disintegrations are measurable measures - when are they continuous?

This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...

**4**

votes

**1**answer

622 views

### Dense inclusions of Banach spaces and their duals

This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, ...

**5**

votes

**3**answers

607 views

### On operator ranges in Hilbert & Banach spaces

Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent:
(1) ran($A$) $\subset$ ...