Questions tagged [norms]
The norms tag has no usage guidance.
339
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For which norms does closest projection never increase norm?
Let $X$ be a finite-dimensional normed space, and $V\subset X$ a vector subspace.
For a point $x\in X$ we define a closest projection of $x$ onto $V$ to be a minimizer $v^\star$ of $\|v-x\|$ amongst $...
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When is an unbounded averaging operator on $\mathbb{R}\to \mathbb{R}$ closed?
Let $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{R}$. Consider the following linear operator $A$ on functions $f:\mathbb{R}\to \mathbb{R}$:
$$(Af)(x) = \sum_{n=1}^\infty a_n f(x+n)+ \sum_{n=1}^\infty a_n f(...
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Small total variation distance between sums of random variables in finite Abelian group implies close to uniform?
Let $\mathbb{G} = \mathbb{Z}/p\mathbb{Z}$ (where $p$ is a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$.
For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\...
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Third roots of unity and norm element
Let $K = \mathbb{Q}(\zeta_3)$ where $\zeta_3$ is a third root of unity, let $F = \mathbb{Q}(\sqrt[3]{\ell})$ where $\ell \equiv 1 \pmod{9}$ is a prime and set $L$ to be the Galois closure of $F$, i.e.,...
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Bounding supremum norm in terms of gradient L2-norm using a Poincare-like inequality
Suppose $f$ is a Lipschitz continuous real-valued function over a bounded domain $\Omega \subset \mathbb{R}^d$ with smooth boundary, and let $\overline{f} := \frac{1}{|\Omega|}\int_\Omega f(x) dx$. Is ...
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Verifying the Cauchy behavior of a sequence
Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
- 75
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On a property for normed spaces
I asked this question on Math Stackexchange, but I didn't get an answer:
https://math.stackexchange.com/questions/4881155/on-a-property-for-normed-spaces?noredirect=1#comment10410489_4881155
I came ...
- 1,361
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Strong sub-differentiability of an equivalent strictly convex norm
First, we define the notion of strong sub-differentiability(SSD) of a norm on a Banach space $X$. The norm $\Vert \cdot \Vert$ of $X$ is said to be SSD if the one-sided limit $$\lim_{t \to 0+} \frac{\...
- 75
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Computing preimage of element under norm map of quadratic extension of $2$-adic fields
Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...
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Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
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Convergence of slice in an equivalent renorming
Let us consider $\ell_2$ space with $\Vert \cdot \Vert_2$ norm. Let us define a new norm equivalent to $\Vert \cdot \Vert_2$ norm as follows:
$$
\Vert x \Vert_0 = \max \{ \Vert x \Vert_2, \sqrt{2} \...
- 75
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Are these two norms on localized versions of $L^p_q$ equivalent?
$\newcommand{\RR}{\mathbb R}\newcommand{\diff}{\, \mathrm d}$ We fix $T \in (0, \infty)$ and $p, q \in [1, \infty)$. Let $\mathbb T$ be the interval $[0, T]$.
Let $E$ be the space of all real-valued ...
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75
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Convexity property of an equivalent norm on $\ell_2$
Let us consider the space $\ell_2$ with an equivalent norm defined by
$$
\Vert x \Vert = \max \{ \Vert x^{'} \Vert_2, \Vert x^{''} \Vert_2 \},
$$
where $x^{'}=(0, x_2, x_3, \cdots)$, $x^{''} = (x_1, 0,...
- 75
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Norm of a matrix function of a vector $x$ [closed]
I have a matrix $$A(x) = \frac{-1}{(1+\|x\|_2^2)^{\frac{3}{2}}}xx^T + \frac{1}{(1+\|x\|_2^2)^{\frac{1}{2}}}I$$ I see that $$\|A(x)\|_2 \le 1 \ \forall x$$
But is $\|A(x)\| \le 1$ in general $\forall x$...
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Does there exists an example of a Banach space that is compactly LUR; but not LUR
We know that a Banach space $X$ is locally uniformly rotund(LUR) if for $x, x_n \in S_X$ with $\Vert x+x_n \Vert \to 2$, we have $x_n \to x$. In the same case, if $(x_n)$ has a convergent subsequence, ...
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