Questions tagged [metric-spaces]
Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space. For questions about Riemannian metrics use the tag (riemannian-geometry) instead.
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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 $p=2$ ...
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Proving that the closure of a set is closed directly
Currently working through Rudin's principle's of mathematical analysis. I am trying to prove directly that the closure of a set is closed but am hitting a wall on one part of the proof. Namely, if we ...
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Infinite-dimensional cube $[0,1]^\mathbb N$ compact/complete under certain metrics?
Consider the infinite-dimensional cube $[0,1]^\mathbb N := \{ (x_i)_{i=1}^\infty \ | \ 0 \le x_i \le 1 \ \forall i \}.$ This space can be equipped with certain metrics. Below are two of them.
$d_\...
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Determine the convexity of a ball in a metric space.
Let $V$ be the set of all Lebesgue integrable functions. $V$ forms a vector space with respect to general function addition and scalar multiplication. Let $X \subset V$ is the set of positive and ...
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Proving that boundedness of a metric space defined in terms of radius and diameters are same without usage of triangle inequality
Diameter boundedness:A metric space $M$ is bounded if for all points $p,q \in M$, we have $d(p,q) <= R$
Radius Boundedness: A metric space is bounded if and only if for every point $p \in M$, there ...
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Two sets having strongly seperated points are strongly seperated themselves
Definition :- Two sets $A$ and $B$ are strongly seperated if there exist open neighbourhoods $U$ and $V$ such that $A \subseteq U , B \subseteq V , U\cap V=\phi$
Question :$A$ and $B$ are two compact ...
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What are the Chebyshev sets for the taxicab metric?
A set of points $S \subseteq \mathbb{R}^n$ is called a Chebyshev set if the metric projection w.r.t $S$ is single-valued. That is, for every point $x\in \mathbb{R}^n$, there is a unique point $y\in S$ ...
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Computing Component of Vector Outside Convex Hull of Set of Vectors
I have a set of points $\{x_n\}_{n=1}^N$ where $x_n \in \mathbb{R}^D$. I have another vector $y \in \mathbb{R}^D$. I'd like to efficiently compute the component of $y$ that lives outside the convex ...
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Always Closed Metric Space is Complete
I am trying to prove that
Given a metric space $Y$, if for every metric space $X \supseteq Y$, $Y$ is $X$-closed, then $Y$ is complete.
by contradiction or contraposition, such that I don't use the ...
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Rudin Ch 4 exercise 3: the zero set of a continuous function is closed
Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$) be the set of all $p \in X$ at which $f(p) = 0$. Prove that $Z(f)$ is closed.
My attempt
Let $p$ be a ...
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Manhattan metric proof
I would like to check my proof of the triangle inequality of the manhattan metric in $\mathbb{R}^2$
i.e: that if $d(x,y) = |x_1 - y_1| + |x_2 - y_2|$ then $d(x,y) \leq d(x,z) + d(z,y)$.
My proof is as ...
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Understanding the Proof of Relative Openness Theorem in Rudin's Principles of Mathematical Analysis
I'm having trouble understanding Rudin's proof for the theorem stating:
"Suppose $Y \subseteq X$. A subset $E$ of $Y$ is open relative to $Y$ if and only if $E = Y \cap G$ for some open subset $G$...
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Are there compact metric spaces with Hausdorff measure equal to 0 or infinty
I am looking for examples of non-empty metric spaces which are compact with Hausdorff dimension $\alpha$, and have either
its $\alpha$-dimensional Hausdorff measure equal to $0$
or
its $\alpha$-...
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A complete metric space contains a convergent sequence or an infinite discrete subset
Theorem. Let $X$ be an infinite complete metric space. Then there is an injective convergent sequence in $X$ or there exists $\varepsilon>0$ and an infinite $\varepsilon$-discrete subset $A\...
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Parametric Equation of a unit circle when the angle between $x$-axis and $y$-axis is not $90$ degrees
I know in regular Cartesian coordinates the parametric equation for a unit circle is $x=\cos(\theta)$, $y=\sin(\theta)$, and if the $x$ coordinates are stretched by an amount $a$, and the $y$ ...
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