All Questions
Tagged with metric-spaces normed-spaces
674
questions
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There is at least one point of every non-empty open subset of the $\ell^2$ space whose first coordinate is nonzero [duplicate]
Here we take
$$
\mathbb{N} := \{ 1, 2, 3, \ldots \}.
$$
Let $\ell^2$ denote the set of all the real (or complex) sequences $\left( \xi_i \right)_{i \in \mathbb{N} }$ such that the series $\sum \left\...
- 26.9k
1
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1
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Convex combination of equidistant curves
Say we have three curves $\gamma, \delta, \varepsilon : \mathbb R \to \mathbb R^n$ such that the distances $\lVert \gamma(t) - \delta(t) \rVert$ and $\lVert \gamma(t) - \varepsilon(t) \rVert$ are ...
- 358
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Which metrics (on vector spaces) can be induced?
Is there a way to classify which metrics defined on vector spaces can be induced by a norm? ie. there exists norm $n: X\to \mathbb{R}$ on the vector space $X$ such that the metric $d(x,y)=n(x-y)$.
I ...
- 477
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If a pseudonorm function $N$ is continuous in a given topology, does the pseudometric topology formed by $d(x,y)=N(x-y)$ coincide with first topology?
Define $p_n\#= p_n p_{n-1} \cdots p_1$ for $n = 0$ to be $1$, then we have a function:
$$
N : \Bbb{Z} \to \Bbb{Z}, \\
N(x) = \left |-1/2 + \sum_{d\ \mid\ p_n\#} (-1)^{\omega d} \sum_{r^2 = 1 \mod d} \...
2
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33
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Linear Analysis – Examples 1-Q5 General $l^p$ spaces vs $l^1, l^\infty$
Let $1<p<\infty$, and let $x$ and $y$ be vectors in $l_p$ with $\left \|x \right \|=\left \|y \right \|=1$ and $\left \|x +y \right \|=2$, how to prove $x=y$?
I know how to prove for $p=2$ ...
- 477
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Does this relation between Wasserstein distances hold: $W_1(\mu,\nu)\leq W_2(\mu,\nu) \leq ... \leq W_\infty(\mu,\nu)$?
I stumbled upon this interesting statement in this paper:
"One interesting observation is that the Wasserstein ambiguity
set with the Wasserstein order p = 2 is less conservative, because the 2-...
- 99
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35
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Proving $g(x) = 1 − \frac{\lVert x−p\rVert}{\delta}$ is continuous
I want to show that given any two points $p, q \in K$ with $p\neq q$ we can choose
a continuous function $g \in C(K)$ so that $g(p)\neq g(q)$, specifically by letting $g(x) = 1 − \frac{\lVert x−p\...
- 477
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14
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Why metric equivalence does not preserve completeness but norm equivalence does? [duplicate]
I was studying Functional analysis when I came across the normed space equivalence i.e. equivalent norms. I already proved that equivalent norms on a vector space $X$ over field $\mathbb{R}$ or $\...
- 23
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Any norm on $\mathbb R^n$ induces product metric?
Let $X_1, \ldots, X_n$ be metric spaces and $\|\cdot\|$ be a norm on $\mathbb R^n$. Then it's easily seen that if $\|\cdot\|$ is monotonic in each coordinate while (others fixed) in the orthant $[0, +\...
- 4,119
1
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2
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39
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Compactness of $C[J,X]$?
If $J=[0,T]\subset \mathbb{R}$, and $X$ is a Banach space? Then, is $C[J,X]$ (Banach space of all continuous functions from $J$ to $X$) a compact metric space with respect to supnorm defined as $\|x\|...
1
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1
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36
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A separable normed space that is continuously embedded in a non-separable normed space implies that this embedding isn't dense.
As a preliminary I introduce the definition of denseness I am using:
Definition (dense subsets of metric spaces). Suppose $(M,d)$ is a metric space. A subset $S \subset M$ is called dense in $M$ if ...
- 1,141
11
votes
1
answer
812
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Can we derive a norm and an inner product from a metric?
Given an inner product on a vector space, I can always define a norm and a metric (and a topology using that metric). Is the converse true? That is, given a metric on a vector space, can I define an ...
- 177
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Is this rewriting of $L_1$ distance on simplex a specific case of a more general concept?
For my use case, it seems that $L_1$ distance on simplex
$$\lVert x - y \rVert_1 = \sum |x_i - y_i|$$
is more useful when written in the following form:
$$\lVert x - y\rVert_1 = 2 - 2\sum \min(x_i, ...
1
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2
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493
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Show that any sequence in $\mathbb{R}^2$ that (...) does not converge to any limit with respect to the sunflower metric.
Working in $\mathbb{R}^2$ with two metrics, the standard (Euclidean) and the sunflower metric:
\begin{equation}
d_{sf}=\begin{cases}
\lVert x-y \rVert & \text{if x and y lie on the same line ...
- 177
3
votes
1
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479
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Quantitative bound on Wasserstein distances by $L^p$ distances?
Given two smooth probability densities $f$ and $g$ on $\mathbb{R}$ (or $\mathbb{R}_+$) with finite $p$-th moments. I am wondering if anyone is aware of some explicit upper bound on $W_p(f,g)$ in terms ...
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