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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Prove Contraction Mapping

The following is given: Eucliden metric $d$, defining the distance between vector $v_1=(x_1,y_1)$ and $v_2=(x_2,y_2)$: $d(v_1,v_2)=\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ $M $ is a mapping of $\mathbb ...
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$C([a,b] \times [c,d],X)$ compared to $C([a,b],C([c,d],X))$ and $C([c,d],C([a,b],X))$

Let $C(Y,X)$ be the space of continuous functions from $Y$ to $X$ together with the supremums norm. Here $Y$ is a compact space and $X$ a metric space. Let $a,b,c,d \in \mathbb R$ be finite, with ...
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29 views

Metric Spaces, Continuity and Preference Relations

Let X be a metric space and $\succeq$ be a preference relation on X. The preference relation is continuous if the sets $\succeq (y) =\{x: x \succeq y\}$ and $\preceq (y) = \{x : x \preceq y\}$ are ...
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25 views

How to prove continuity of addition over weird metric? Edit: Ignore this. Errors in the problem definition.

Let $f: R \times R \rightarrow R$ and let the metric over $R$ be $d(x,y)=|x-y|$ and let the metric in $R \times R$ be $d_2((x,y),(a,b))= ((x-y)^2+(a-b)^2)^{1/2}$. I believe I understand how to ...
0
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1answer
19 views

Finding open balls in R2

If anyone can help I would be highly grateful! The Problem is in the image below.. [1] http://i.stack.imgur.com/eKDH2.png Should you approach using the open balls to find the boundary of the set? ...
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1answer
13 views

Drawing Ball (0,1) in half Euclidean Metric [closed]

Just a quick question. I know it will be a circle of radius 2, but can somebody just clarify why?
3
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69 views

shortest path in complete metric space

Let $(X,d)$ be a complete connected by arcs metric space. We define the length of a continuous path $\gamma: [0,1] \rightarrow X$ to be \begin{equation*} \sup\limits_{0=a_{0}<a_{1}<... a_{n}=1} ...
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1answer
20 views

When are the following inclusions $\subsetneq$

When does the "equality" part of inclusion fail in: $$\overline{A \cap B} \subseteq \overline{A} \cap \overline{B}$$ and $$Int(A \cup B) \supseteq Int(A) \cup Int(B)$$ ? Can you provide an simple ...
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22 views

continuity of a metric d

from Continuity of the Metric and Convergence Sequences, why $d^{-1}(V)$ is an open ball? to be an open ball, I think it contains elements of $X$, not $X^{2}$. why is it?
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Showing that $f$ continuous

Let $A$ be a compact subset of a metric space $(X, d)$. Consider the function $f : X → R$ given by $f(x) := $ sup $ \{d(x, y) : y ∈ A\}$ . Show that $f$ is continuous. I tried taking an open subspace ...
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$ {\|f\|}_p = \sqrt[p]{\int_{a}^{b} |f(x)|^p {\rm d}x}$ is a norm

Consider the space $C([a,b])$ of all continuous functions $f\colon [a,b]\rightarrow \mathbb{R}.$ Show that the function $\|\cdot\|_p\colon C([a,b]) \rightarrow [0,\infty),p>1$, given by $$ ...
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4answers
31 views

Prove $\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2$

If $X,Y$ are vectors in $\mathbb{R}^n$ and $a>0$ show that: $$\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2 (*)$$ I started with ...
0
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1answer
13 views

Minimal conditions for $\widetilde{d}$ to be metric

Let $(X,d)$ be an arbitrary metric space and $f:[0,\infty) \rightarrow [0,\infty)$ What are the minimal conditions for function $f$ in order $\widetilde{d} = f \circ d: X \times X \rightarrow ...
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24 views

Is $\phi$ a norm of E?

Let $(E, \| \|)$ be a normed space. We define $\phi:E \rightarrow [0,\infty)$ as follows: $$\phi(e)= \dfrac{\|e\|}{1+\|e\|}$$ Is $\phi$ a norm of $E$? Please help! Thank you! P.S. This question ...
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Non-Banach, completely metrizable normed vector space

Does there exist a normed vector space $(X,\|\cdot\|)$ over $\mathbb R$ or $\mathbb C$ such that the metric induced by the norm $(x,y)\mapsto\|x-y\|$ is not complete; but there exists some other ...
3
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If $\inf \{ d(x,y)\mid y \in C \}=0$, then $d(x,z_n)< \frac{1}{n}$

I'm studying a proof I learned in class and I don't quite understand this statement. Let $X$ be a metric space and $C \subset Z$ a nonempty closed set. For each $x \in X$ define $f_{c}(x)=$ inf ...
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49 views

Lebesgue number lemma fails for the plane

I need to show that Lebesgue number lemma fails for the plane. I am clueless how to show this. Lemma : Let $(X, d)$ be a compact metric space. Then given an open cover $\mathcal{A}$ of $X$, ...
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4answers
65 views

Does every compact metric space contain a sequence such that every point of the space is a limit point? [closed]

Let $(X,d)$ a compact metric space. Now, I would like to know if there exists a sequence such that each point in $X$ is a Limit point of this sequence?
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11 views

Formal proof that diameter of subset is bounded by diameter of superset (in metric space)

I am asked to show that if $A \subset B$, then $\delta(A)\leq \delta(B)$, where $\delta(A)=\sup_{x,y \in A}d(x,y)$ is the diameter for the non-empty set A in the metric space $(X,d)$. The fact that ...
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Proving a complete and totally bounded metric space is compact.

I'm having trouble writing down the details of this proof formally. Statement: Suppose $(X, d)$ is a metric space that is complete, and totally bounded (i.e., for every $\epsilon > 0$, ...
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27 views

Relation between connected subset and clopen subset of a metric space?

I've read that for $A$ a connected subset of a metric space $M$ and $C$ a clopen (closed and open) subset of $M$, one could prove that either $A \subset C$ or $A \cap C=\varnothing$ and use it to ...
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Sequence, subsequences, sub-subsequences in metric spaces

Let $(X,d)$ be a metric space, and let $\{x_n\}_{n \in \mathbb{N}}$ be a sequence in $X$. Assume that every subsequence of $\{x_n\}_{n \in \mathbb{N}}$ has a sub-subsequence that converges to the same ...
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1answer
33 views

Calculate X Y Z from two specific degrees on a sphere

I am a programmer, don't know much about advanced math. I would need the exact formula(s) that could achieve this, so I can translate it to my programming language. I am having a headache trying to ...
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1answer
66 views

A property of compact subsets of metric spaces

Let $(X,\varrho)$ be a metric space and $K\subset X$ compact. Then, for every $\,\varepsilon > 0$, $\,K$ can be covered with a finite number of balls of radius $\varepsilon$. Show that the ...
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Defining metrics as a function to something other than the reals.

Generally speaking, a metric for a space R is defined as a function from RxR -> Reals, but does it have to be? Can we define it in more generic terms such as a function from R to a field with certain ...
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Is $A$ compact, $f(A)$ uniformly continuous and is $f^{-1}$ continuous?

$X$ and $Y$ are metric spaces, $A\subseteq X$, $A$ is bounded. map $f:X\to Y$ is continuous. Questions: Is $A$ necessarily compact? Is $f(A)$ uniformly continuous? If given that $f$ is a ...
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How do you prove that Z (the set of integers) equipped with the Euclidean metric (induced from real numbers) is complete?

How do you prove that $\mathbb{Z}$ (the set of integers) equipped with the Euclidean metric (induced from real numbers) is complete? I am having trouble with this question, I don't really know where ...
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On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
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How to define distance between two functions in a non-linear space (example of non-linear space: shape space)?

Suppose I have two parametric circle $f_1=(acost,asint)$ and $f_2=(bcos t,bsint)$, $t\in(0,2\pi),a>0,b>0$, which lies in some non-linear space. Are there any way, how to define the ...
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When a metric space is a normed space?

I'm trying to figure out that which condition should be provided for a metric space to be normed also?
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If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable

I need to show that: If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable. I have already showed that every locally compact Hausdorff space ...
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3answers
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Prove the set, {y ∈ X | r ≤ d(x,y) ≤ s}, is closed

Let r < s be positive real numbers and x ∈ X. Prove that the set: {y ∈ X | r ≤ d(x,y) ≤ s}, is closed. Having trouble with how I should tackle this ...
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1answer
60 views

Retraction to an interval in a metric space

Suppose that $X$ is a metric space and $A$ is a subspace of $X$ that is homeomorphic to the interval $[0,1]$ with its usual topology. Let $v$ and end point of A. How do you proof that there is a ...
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1answer
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A Base of a metric space intuition

From what I have read online and from what I have read in Rudin, a collection of open sets $\lbrace$$V_{n}$$\rbrace$ is said to be a base for a metric space $X$ if every open set in $X$ can be ...
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Cantor's intersection Theorem without the diameter hypothesis

In proving Cantor's in intersection theorem, the fact that limit of the diameter of the sets is 0 was used to prove that the intersection is non-empty. I just wondered if that hypothesis is excluded ...
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21 views

Calculate projection of a line in a square

Said that we have two points (P1, P2) that form a line, and 3 points (S1,S2,S3) that form a square, how would we calculate the position X and Y of the point resulting from the intersection of the line ...
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Topology of metric completion of Euclidean metric

Lets consider $\cal{M}=\mathbb{R}^{2}\backslash\{(0,y)\}\text { with } \{|y|\le1\}$ with the Euclidean metric with line element $ds^{2}=dx^{2}+dy^{2}$. Now consider the distance function given by ...
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1answer
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Problem about completeness

Does there exist a complete metric on $(0,1)$ inducing the usual topology? My problem is that I cant understand what will I have to do to answer the question.It's a problem of a competitive exam.
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1answer
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Showing the right half of the unit hyperbola is a complete metric space.

Let $f : \mathbb{R} \rightarrow \mathbb{R}^2$ be given as follows. $$f(\theta) = (\cosh \theta, \sinh \theta)$$ I want to argue that $\mathrm{im}(f)$ is a complete metric space with respect to the ...
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Normed Space and Hibert Space Problem

Anyone could describe me, why this is True? Suppose $(H, \|.\|) $ is a normed space. the norm $\|.\|$ induced by an inner product if and only if Parallelogram law is valid. Regards.
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Example of $x$ being adherent point but not accumulation point?

So I was just reading Apostol and I see that if $x$ is an accumulation point of set $S$, it has to be an adherent point as well. I guess it's possible for $x$ to be an adherent point only, not an ...
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Let $(X,d)$ be a metric space, $C$ a compact subset, and $K$ a closed subset. Prove that $K \cap C = \emptyset$ iff $d(K,C) > 0$.

Let $(X,d)$ be a metric space, $C$ a compact subset, and $K$ a closed subset. Prove that $K \cap C = \emptyset$ iff $d(K,C) > 0$. For this problem I was going to consider $d(x,F) = \inf d(x,y)$ ...
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0answers
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How to show that addition is continuous?

Let $f: R \times R \rightarrow R$ and let the metric over $R$ be $d(x,y)=|x-y|$ and let the metric in $R \times R$ be $d_2((x,y),(a,b))= ((x-y)^2+(a-b)^2)^{1/2}$. I believe I understand how to ...
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Is every connected subset of the Sierpiński triangle arcwise connected?

I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome. I managed to prove that (assuming $S$ is the connected subset) $S$ ...
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Convergence of functions in a metric space

Let $C([0,1])$ be the space of all continuous functions from $[0,1]$ to $\mathbb{R}$ under the metric $$ \lVert f \rVert_1 = \int_0^1 \lvert f(x) \rvert \, dx. $$ Now consider $f_n(x) = e^{-nx}$. I ...
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Proving a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
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Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true?

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true for all $x,y\in\mathbb{R}$? If not, how can I prove that $\int\frac{\vert ...
0
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1answer
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Closed Interval in $E^2$

I am currently working through Introduction to Analysis by Rosenlicht In one of the exercises $4.30,$ he asks a question regarding a closed interval in $E^2.$ I am not sure what this means. I was ...
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1answer
31 views

Any open set shares boundary with a discrete set

Claim: Let $X$ be a metric space and let $U\subset X$ be open. Then there exists a discrete set $A\subset X$ such that $\partial A = \partial U$. Approach thus far: Since this statement is about ...
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0answers
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Problem in proving norm

I have a linear space V that includes the continuous functions from [-$\pi, \pi$] to the complex set C, of the form: $$f(x) = a\cos(t)+b\sin(t) $$ where $a$ and $b$ are complex numbers. I want to ...