# 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.

10,282 questions
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### Complete space of functions with $f(a)=f(b)$?

Consider M is space of continuous functions (on $[a,b]$) with condition: $f(a)=f(b)$ . Is it complete metric space with $\mu(f,g) =\max\underset{x\in [a,b]}{|(f(x)-g(x))|}$? In my opinion it's true. ...
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### Homeomorphism between $([0,1]^2, \delta )$ and $[0,1]^2$ with standard euclidean metric

I am stuck as to how I find the homeomorphism described above. $\delta$ is here described to be the metric $$\delta((m_1,n_1), (m_2,n_2))= max\{d_M(m_1,m_2),d_N(n_1,n_2)\}$$ a metric on $M\times N$ ...
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### How do I visualize a set in metric space?

If I'm given a metric, say the discrete metric $\text d_0(x,y):=\begin{cases} 0, & \text{if }\vec x= \vec y\\ 1, & \text{if } \vec x \neq \vec y\ \end{cases}$ and want to visualize a set ...
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### Equality of certain distances in a normed vector subspace

Let $E$ be a normed vector space and $F$ a vector subspace of $E$. If $y \in F$, $x \in E$ and $0 < a \in \mathbb R$, prove that $d(y + ax, F) = ad(x,F)$. I've tried to write down the definitions ...
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### Metric on Lorentzian manifold

Since the signature of a Lorentzian manifold (M,g) is (-,+,+,+), am I right to assume that the determinant of $g_{ij}$ with EACH coordinate system is $-1$?
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### A sequence of continuously differentiable functions with non-differentiable limit

Working in the metric space $C[a,b]$, the set of continuous functions $f:[a,b]\rightarrow[a,b]$, with the supremum metric, I need to demonstrate that $C^1[a,b]$, the subset of continuously ...
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### A set $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ whenever $z \in \Omega$. The set $\Omega$ is contained in some large disc.

My complex analysis textbook says the following: A set $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ whenever $z \in \Omega$. In other words, the set $\Omega$ is contained ...
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### Define a metric on collection of compacts

Let $(\mathbb{R}^2,d)$ be a metric space, where $d$ is the Euclidean metric on $\mathbb{R}^2$. Let $\kappa$ denote the set of compact subsets of $R^2$. Which one of the following expression defines a ...
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### Relationship between Riemann metric and divergence in information geometry

I would like to ask you about formula(6) in this paper. Some people say that the Primary term of the divergence $D$ disappear. But how should I show that?
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### Uniqueness of completion of a metric space

I've been reading Aliprantis and Burkinshaw. This book gives a nicer construction of completion of metric space(in my opinion) avoiding any equivalence class or pseudometrics. It gives a simple ...
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### Contraction mapping on metric space

We know the definition of contraction mapping. But it is unkown to me the definition of weak contraction mapping. Help me
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### Which of the following statements are true?(NBHM-2016,Topology(3.7))

My Try (a)$K\subset M_n(\mathbb R)\sim \mathbb R^{n^2}$ compact iff $K$ is closed and bounded. $A\in K$ has entries from the bounded set. So, eigenvalues are bounded. (b) $tr(A)=1$ is a hyperplane ...
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### Boundary Points and Metric space

Definition:The boundary of a subset of a metric space X is defined to be the set $\partial{E}$ $=$ $\bar{E} \cap \overline{X\setminus E}$ Definition: A subset E of X is closed if it is equal to its ...
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### Open sets in $\mathbb R^2$ and $\mathbb E^2$

Can someone help me with the following problem: Prove that every open set in $\mathbb E^2$ is also open in $(\mathbb R^2, d_1)$, where $d_1((x_1,y_1),(x_2, y_2))=|x_1-x_2|+|y_1-y_2|$ , and vise versa,...
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