# Tagged Questions

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### Show the following is Cauchy:

I am trying to prove that the Euclidean Norm/inner product on C([0,1]) does not give rise to a complete metric space. To do this I am trying to find a Cauchy Sequence which does not converge in ...
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### (i) Show that T is continuous on $(X,d)$. (ii) Show that T is continuous on $(X,d_{2})$.

Let $K(t,s)$ be a continuous function on $[0,1]\times{[0,1]}$. Let $X=C[0,1]$ be the set of continuous functions defined on the interval $[0,1]$. Define the mapping $T:X\rightarrow{X}$ by: for every ...
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### Conditions to make a function a metric on $\mathbb{R}$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$. What conditions ensure that $d(x,y)=|f(x)-f(y)|$ defines a metric on $\mathbb{R}$ Let $g:[0,\infty) \to \mathbb{R}$. What conditions on $g$ ensure that ...
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### Is this claim harder to prove for arbitrary metric spaces than for the reals?

Let $X$ and $Y$ be metric spaces. Define the distance between functions $f, g$ from $X$ to $Y$ as $$d(f, g) = \sup_{x \in X} \frac{d(f(x), g(x))}{1+d(f(x), g(x))}$$ Is it true that if $f_n:X \to Y$ ...
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### Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$

Let $(X,d)$ be a metric space. Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$ such that $E\cap A\neq\emptyset, E\cap B\neq \emptyset$ and $E\subset A\cup B$. ...
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### Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
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### Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
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### Open sets are $\mu*$-measurable in a metric space with a given condition

I have a homework problem that I'm very stuck on. The problem statement is as follows: "Suppose that $X$ is a metric space, and that for any sets $E,F \subseteq X$, if dist$(E,F) > 0$ then ...
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### Pyramid Question

Let us suppose we have a right pyramid,that is a pyramid with all edges equal to each other and a height which goes to the center of the geometric shape that serves as it's base. We than also know ...
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### Given a metric space $(X,\rho)$, prove that $|\rho(x,z)-\rho(y,u)|\leq{\rho(x,y)+\rho(z,u)}$ for $x, y, z, u\in{X}$.

Obviously it is true, but I'm not sure how to prove it. I'm considering the quadrilateral inequality but so far it has not been helpful. Can anyone give me direction on how to verify ...
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### Is there a non-complete and non-separate metric space?

Is there a (non-trival) non-complete and non-separate metric space? Some notions are here: math.stackexchange.com/questions/182316.
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### Let $a,b \in \mathbb{Z}$ with $a<b$. Determine $d_n:= | \{ c\in\frac{1}{n}\mathbb{Z} \ | \ a < c < b \} |$

The assignment is: Let $a,b \in \mathbb{Z}$ with $a<b$. For $n\in\mathbb{N}$ determine $d_n:= | \{ c\in\frac{1}{n}\mathbb{Z} \ | \ a < c < b \} |$. Firstly, I began to see what the ...
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### Hadamard space: property of the Busemann function

I have a question about a property of Busemann functions on Hadamard spaces. Let $X$ be a complete CAT($0$) space. If $r:[0, \infty) \to X$ is a geodesic ray, and $x\in X$ the Busemann function is ...
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### Is $\rho(x,y)=(x-y)^2$, with $x,y\in \mathbb{R}^1$, a metric space on $\mathbb{R}^1$?

Obviously it has to satisfy the following: 1) For all $x,y\in X$, $0\le d(x,y)$. (positivity) 2) For all $x,y\in X$, $d(x,y)=d(y,x)$. (symmetry) 3) For all $x,y,z\in X$, $d(x,y)\le d(x,z)+d(z,y)$. ...
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### Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
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### Show that $d^\ast$ is a metric

For $x$ and $y$ in $R$, let $d(x,y)$ be a metric. Show that $$d^\ast(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ is also a metric. It is fairly straightforward to show that $d^\ast(x,y)=0$ if $x=y$ ...
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### Please help me prove $K \cap A' \neq \varnothing$

I need prove the follow example. Please help me: The example is: If $A$ is an infinite subset of a compact set $K$, then $K \cap A' \neq \varnothing$. Definition: Point $x\in X$ is a ...
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### Continuity and sequential continuity

Prove that: > The function $f:(X,d)\rightarrow(Y,\rho)$ is continuous if and only if $f$ is sequentially continuous (that means $x_n\rightarrow x \Rightarrow f(x_n)\rightarrow f(x)$) Proof. ...
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### Show that $\Bbb Q$ and $\Bbb R \backslash \Bbb Q$ with restriction for euclid metrics is not is not complete

Please someone help me because I can not solve the following example. Show that $\Bbb Q$ and $\Bbb R \backslash \Bbb Q$ with restriction for euclid metrics is not is not complete. Plase help me. ...
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### Diameter of a subset of a metric space

Let $(\Bbb R,d)$ be the metric space with the metric function $$d(x,y)=\frac{|x-y|}{1 + |x-y|}\;.$$ Calculate $\operatorname{diam}(0,\infty)$. I am thinking the answer is $1$ because ...
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### If $\{x\}$ is an open set in $X$, for all $x\in X$, then all subsets of $X$ is open in $X$

Today for example the teacher ask us to nejdeme next example, but none of us knew, so we left for example homework, but try again but I can not solve the example, so please someone help me, the ...
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### Metric space question

Let (M,p) be a metric space and suppose that ${x_n}$ is a sequence in (M,p) so that $x_n -> x$ and $x_n->y$. prove x=y Let $E>0$. then, $p(x_n,x)->0$ $lim$ $n->inf$ ...
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### Proving various properties of metric spaces

Suppose that $p_1$ and $p_2$ are metrics on $M$. Prove that the following are also metrics: (a) $p = p_1 + p_2$ define $p_1(x,y) = |x-y|$ define $p_2(a,b) = |a-b|. p = p_1+p_2 = |x-y|+|a-b|$. But I ...
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### Homeomorphisms of disjoint unions and unions in a metric space.

Suppose $A$ and $B$ are disjoint subsets of a metric space $X$, equip $A$, $B$, and $A\cup B$ with the subspace topology. Suppose $d(x,y)\geq \delta$ for all $x\in A, y\in B$ and some $\delta>0$. I ...