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$L^1$ convergence

Setting For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
Anthony's user avatar
  • 125
0 votes
1 answer
242 views

When do the weak-star and compact convergence (compact-open) topology coincide on the dual of a Banach space?

In Measures Which Agree on Balls by Hoffmann-Jørgenson, it is claimed on page 323 that for an arbitrary Banach space $E$, if $\pi$ is the topology on $E^*$ of uniform convergence on compact subsets of ...
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0 answers
94 views

When can an affine functional on the dual be represented as an element of a Banach space?

In Measures Which Agree on Balls by Hoffmann-Jørgenson, we are given a functional $\varphi: T(x_0)\to (-\infty, \infty]$, which is a lower semicontinuous, affine, Baire function on a subspace $T(x_0)$ ...
i like math's user avatar
1 vote
2 answers
115 views

Computation of tangent functional

In Measures Which Agree on Balls by Hoffmann-Jørgenson, the tangent functional is defined as follows. If $x \in S$, we define the tangent functional $\tau(x,\cdot)$ at $x$ as \begin{equation} \...
i like math's user avatar
0 votes
0 answers
127 views

Closure of BV paths in space of paths of finite $p$-variation

Let $p\ge1$ and $T>0$. Define $\mathscr D([0,T])$to be the space of partitions of $[0,T]$, where each partition is a finite collection of distinct points of $[0,T]$. Consider a continuous path $X:[...
Martin Geller's user avatar
1 vote
1 answer
163 views

Complex interpolation of subspaces

Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\...
A beginner mathmatician's user avatar
2 votes
2 answers
176 views

Direct limit of the sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category of Banach spaces

Recently I have been reading the paper The categorical origins of Lebesgue integration by Tom Leinster (https://arxiv.org/pdf/2011.00412.pdf). In this paper, he said that: For $n \geq 0$, let $E_{n}$ ...
ScienceAge's user avatar
0 votes
1 answer
218 views

Heat semigroup dissipative

Consider the heat semigroup on $L^1(\mathbb{R}).$ I would like to know if the generator of this semigroup is dissipative in the sense of this definition. On $L^2$ it would be completely trivial, but ...
Zehner's user avatar
  • 167
3 votes
1 answer
255 views

A differentiable version of the Michael selection theorem

Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map. Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?
Ali Taghavi's user avatar
3 votes
1 answer
431 views

A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable

Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet ...
Ali Taghavi's user avatar
5 votes
1 answer
794 views

Can the Sobolev norm of order 1/2 detect "jumps"?

We are given a function $f: \mathbb R^d \to \mathbb R$. For simplicity we can assume that $f$ is smooth and compactly supported. Is the Sobolev norm of order $\frac{1}{2}$ strong enough to prove an ...
Martins Bruveris's user avatar
2 votes
2 answers
354 views

A bound on linear functionals over cotype 2 spaces

This is a modification of the somewhat naive question that I asked below. Suppose $X$ is a real Banach space of cotype-2, and $u_1, u_2, ... u_n$ are unit vectors in this space. For $\gamma = ((\...
Brad Rodgers's user avatar
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