Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Proof that boundedness of continuous Real Valued functions implies Compactness

I'm looking to prove the following : Let $(X,d)$ be a Metric Space If every continuous real-valued function on $X$ is bounded then $X$ is Compact I saw a proof earlier today If instead $X$ is ...
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Completion of the real numbers

On the real line $\mathbb{R}$ endowed with euclidean topology i may put different metrics, inducing the same topology, but inducing different completions. For example if one considers the standard ...
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True/False? If $a ∈ iso(S)$ , then, $a_i ∈ iso(π_i(S))$ for all $i ∈ \mathbb N_n,$ where $π_i$ denotes the natural projection of $P$ onto $X_i$

Suppose $n ∈ \mathbb N$ and, for each $i ∈ \mathbb N_n, (X_i, τ_i)$ is a metric space. Suppose $d$ is a conserving metric on $P = \prod_{i=1} ^n X_i .$ Suppose $S ⊆ P$ and $a ∈ S.$ Is it true that If ...
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Distance between sets

Let $K \subset K_1 \subset U \subset \Bbb R^2$, such that $K$ and $K_1$ are compact sets, with $K \subset \mathring {K_1}$, and $U = \mathring U \subsetneq \Bbb R ^2$. If $w \in \partial K_1$ such ...
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Proposed proof for quasi-metric result

A quasi-metric on a set $X$ is mapping $\rho: X \times X \rightarrow [0, \infty)$ satisfying the following conditions: $\rho(x,y) \geq 0~~\text{and}~~\rho(x,x) = 0;$ $\rho(x,z) \leq \rho(x,y) + ...
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23 views

Series Convergence in Banach Space

Let $(e_j)_1^\infty$ be an orthonormal set in $l^2$ Consider $$s_n =\sum_{j=1}^n t_je_j$$ Show that $s_n$ converges in $l^2 \iff t = (t_j)_{j=1}^\infty \in l^2$ Thoughts so far : If we consider ...
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Outer Regularity of the Lebesgue measure on the Hilbert brick

Is the product measure on the Hilbert brick $I=[0,1]^\mathbb{N}$ outer regular (that is measure of every set is the inf of measures of open sets, containing it)?
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Metric for connected path space.

I'm trying to prove the next function is a metric for the space of connected paths $T_{x,y}(X)$ where $x,y\in X\subset\mathbb{R}^{n}:$ $$d(x,y)=\inf\{L(\sigma):\sigma\in T_{x,y}(X)\},$$ where ...
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limit points of subset of real numbers

Let $$A=\{ \frac{\sqrt{m} -\sqrt{n}}{\sqrt{m}+\sqrt{n}} | m,n\in \Bbb{N} \}$$ I think that we must find sequence of $A$ and find limit of sequence,let $a_m =\frac{\sqrt{ k ^2 m^2} -\sqrt{ ...
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Ball with euclidean metric - mistake in book?

In $\mathbb{R}^2$, the ball with euclidean metric $d_{l_2}$ is defined, in terence tao's analysis vol II, as: $$B_{(\mathbb{R}^2,d_{l_2})}((0,0),1) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 <1\}$$ ...
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Show that $ℓ_2(X)$ is Hilbert space for every set $X$

Show that $ℓ_2(X)$ is Hilbert space for every set $X$ I tryed to find a proof for this problem but i couldn't (searched on internet and mathematical books.Can we find a completed proof for this?
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41 views

Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space) "Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed ...
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71 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
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1answer
52 views

Why is $f(x,y)$ said to be discontinuous at $(0,0)$?

Why is $f(x,y)=\begin{cases} \frac{x^2y}{x^4+y^2}, & \text{if $(x,y)\neq (0,0)$}\\[2ex] 0, & \text{if $(x,y)=(0,0)$} \end{cases}$ said to be discontinuous at $(0,0)$? I am supposed to show ...
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3answers
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If $K\in \Bbb{R}^n$ is compact, $\sup_{x,y\in K}|x-y|=\max_{x,y\in K}|x-y|$.

Suppose $K\in \Bbb{R}^n$ is compact. Let us denote $D=\sup_{x,y\in K}|x-y|$ as $K's$ diameter. Prove there exist $a,b\in K$ such that $D=|a-b|$ i.e, that the suprimum is the maximum. I know there ...
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41 views

Show that the space $ℓ^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0\text{ for } j>>1\}$ is not complete

Show that the space $$\ell^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0 \text{ for } j\gg1\}$$ with inner product $$(a,b) \in ℓ^0\timesℓ^0 \mapsto \langle a,b\rangle =\sum_{j=1}^\infty ...
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1answer
27 views

Uniform convergence on an interval.

Let $f_n(x)=\frac{x^n}{1+x^n},~x\geq 0, ~n\in \mathbb{N}$ Show that there is no uniform convergence on $[1,+\infty[$. I found this particular part of an exercise in my textbook and ...
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Differentiating $f(x)=\sum_{i=1}^{N}|x-y_i|^2$ where $y_1,…,y_N\in \Bbb{R}^n$.

Let $y_1,...,y_N\in \Bbb{R}^n$ and let $f(x)=\sum_{i=1}^{N}|x-y_i|^2$. I need to show that $f$ has a minimum. I try to differentiate but I am having troubles doing so. First of all, does $|x-y_i|$ ...
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1answer
25 views

Distance of a point to a subset.

Let $(M,d)$ be a metric space. For a subset $A\subseteq M$ we define the distance of a point $x$ to $A$ as $$\alpha_A(x):=\operatorname{dist}(x,A):=\inf_{y\in A}d(x,y)$$ Prove that: ...
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Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
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36 views

Does the zeros of an analytic function $f$ form a discrete set? [duplicate]

A function $f$ is called analytic if locally it is given by a convergent power series. Let $ U \subset \mathbb R^n $ be an open set and $f : U \to \mathbb R$ be non zero analytic function. Does the ...
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1answer
36 views

If a set $E\subset \Bbb{R}^n$ is closed and open, then it is either $\Bbb{R}^n$ or $\emptyset$.

If a set $E\subset \Bbb{R}^n$ is closed and open, then it is either $\Bbb{R}^n$ or $\emptyset$. I have an attempt. I know, or at least think, that it is correct ideally, but I don't know how to make ...
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2answers
18 views

Is homeomorphic image of closed bounded subsets of metric spaces , also closed bounded in the homeomorphic image metric space?

Let $X$ , $Y$ be homeomorphic metric spaces with homeomorphism $f$ , then is it true that for any closed bounded subset $A$ of $X$ , $f(A)$ is also closed and bounded in $Y$ ?
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Suppose $S$ is a subset of $\mathbb R$ and $x \in \mathbb R$. Show that there are at most two nearest points of $S$ to $x$

Suppose $S$ is a subset of $\mathbb R$ and $x \in \mathbb R$. Show that there are at most two nearest points of $S$ to $x$ Attempt: Suppose there are three nearest points $a,b,c$ of $S$ to $x$. Let ...
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How can I show uniform convergence?

Let $f_n(x)=\frac{x^n}{1+x^n},~x\geq 0, ~n\in\mathbb{N}$. 1.1.: Determine the pointwise limi of $(f_n)$, $x\geq 0$. 1.2.: Show that the sequence $(f_n)$ is uniformly convergent on ...
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1answer
35 views

Uniform continuity.

Check if the mappings $\mathbb{R}\to\mathbb{R},x\mapsto x^2$ and $[0,\infty[:\mathbb{R},x\mapsto \sqrt{x}$ are uniformly continuous. I was going through some old exams our teacher ...
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32 views

Contraction on a metric.

Let $M=\{x\in \mathbb{R}|x\geq 1\}$ with the absolute metric, being a metric space. Show that: a) The mapping $f:M\to M$ with $f:M\to M,~f(x)=\frac{1}{x}+\frac{x}{2}$ is a ...
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35 views

A ball with respect to a discrete metric

Suppose we have a ball of radius $1$, then: $$B_{\mathbb{R}^2,d_{disc}}((0,0),1) = \{(0,0)\}$$ But if we increase the radius to be larger than $1$, how is it that the ball would encompass all of ...
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What logical resources does one need to distinguish $\mathbb{R}\hspace{-0.05 in}-\hspace{-0.04 in}\mathbb{Q}$ from $\mathbb{Q}$?

(I'm inspired by this question.) As a strict lower bound, having just the order relation is not enough, since $\mathbb{R}\hspace{-0.05 in}-\hspace{-0.04 in}\mathbb{Q}$ and $\mathbb{Q}$ are both ...
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44 views

No convergence in Discrete metric - why?

If I have a sequence $(x_n)_{n \in \mathbb{N}} := (\cfrac{1}{n},\cfrac{1}{n})$ in $\mathbb{R}^2$ Then why isn't there convergence with respect to the Discrete metric? for a discrete metric, the ...
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1answer
83 views

Rare metric space proof [on hold]

Let $C = \mathbb R^{\mathbb N}$ be the space of all possible real sequences. Let's define a metric for this space: $$ d((x_1, x_2, \ldots),(y_1, y_2, \ldots)) = \begin{cases} ...
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1answer
44 views

Metric Space Definition

From my book, the definition given is: Given a set $X$, a function $d: X \times X \to \mathbb{R}$ is a metric on $X$ if for all $x,y \in X \dots$ Then a metric space is a set $X$ together with a ...
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Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, ...
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38 views

Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
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Discrete metric means all sets are countable?

I was working on a proof of "Show that if $A \subseteq \Re^2$ is discrete, then A is a countable set." and I thought about using the discrete metric ($d(x,y)=\delta_{xy}$) on the set as an example ...
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Computing the Log-Euclidean distance efficiently by using eigen-analysis.

Let $A,B\in\Bbb{S}_{++}^n$ be two symmetric positive definite $n\times n$ matrices with real entries. The Log-Euclidean distance between these matrices is defined as follows $$ d = \lVert \log(A) - ...
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80 views

What does it mean for a number to be “larger” than another number? [closed]

In everyday life we have a clear notion of what it means for something to be larger than something else. Usually we would evaluate something's size based on it's volume. However, is there a formal ...
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1answer
38 views

How does this imply “the unit balls completely determine the metric space structure”?

Let $X$ be a metric space. Suppose you know that $$B[x,r] = x + rB[0,1]\qquad\text{and}\qquad B(x,r) = x + rB(0,1)$$ where $B[x,r]$ and $B(x,r)$ are respectively the closed and open balls in $X$. How ...
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Existence of a bounded ball

Lets define: $$F(X) :=\{A \subseteq X \mid A \neq \emptyset , A = \overline{A}\}.$$ For $A, B \in F(X)$ and $p \in X$ define $$d_p(A,B) = \sup_{x \in X} \{ | \operatorname{dist}(x,A) - ...
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Comparing geodesics on a hypersphere

Background: In Euclidean space, a simple and easy to compute distance metric between two vectors $\mathbf{u}$ and $\mathbf{v}$, is the cosine similarity $ \mathbf{u} \cdot \mathbf{v} / |\mathbf{u}| ...
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3answers
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Limit in the set of real sequences.

I have troubles trying to prove the following proposition: Let $S$ be the set of real sequences with ...
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38 views

A question about discrete topological space [closed]

Let $(X,τ_δ)$ discrete topological space i have a question about it Is it locally compact space or not ?
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42 views

Why are all open subsets not infinite in extent?

I have been looking at the definition of an open set, for a metric space. I have come across the following definition, a few times: An open set $U$ of the metric space $(X,d)$ is a set given that ...
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Example of a metric group over $\mathbb{R}_0^+$

Do you know an example of a function $d:\mathbb{R}_0^+\times\mathbb{R}_0^+\to \mathbb{R}_0^+$ for which the following properties hold? Or can you prove this does not exist? There exists an $e\in ...
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Cauchy sequences of two metric spaces with only different distance functions

Consider two metric spaces $(X,d)$ and $(X,d_1)$. If given that for a fixed natural number $N$ we have $$\large{d_1(x,y) = \frac{d(x,y)}{N}, \forall x,y \in X}$$, is there any relation between the ...
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21 views

Examples of space which is not totally bounded

I know some examples of spaces which are not totally bounded. For example, the real space $R$ with discrete metric is bounded but not totally bounded. I understand its not totally bounded because the ...
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1answer
66 views

Is this thing a value quantale?

I am currently trying to understand R. C. Flagg's "Quantales and continuity spaces". However I am struggling a bit with his definitions and would like to have a good simple (but not too simple) ...
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1answer
49 views

Metric on Homogeneous Space $G/H$

For simplicity, assume $G$ is compact and semi-simple Lie group, and $H$ is a closed subgroup of $G$. Therefore the homogeneous space is reductvie, say $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ where ...
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1answer
36 views

Prove completeness of a metric space

Let $\mathcal{K} = \{A \subset \mathbb{R}^N| A \neq \emptyset, A \text{ closed and bounded with respect to the euclidean metric} \}$ Let us define $A_\epsilon = \bigcup_{x \in A}U_\epsilon(x)$, where ...
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41 views

Show that $X-C$ is connected [duplicate]

Let $X$ be a connected metric space, let $A \subseteq X$ be a connected set. Let $\mathcal{C}$ be a connected component of $X-A$. Show that $X-\mathcal{C}$ is connected. Ok, so Im been dealing ...