Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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Prove that the given set is a metric space?

Prove that if $d$ is a metric and $(X,d)$ is a metric space then $D(x,y)=(d(x,y))^2$ is also a metric on $X$. I have problem of showing that $D(x,y)$ is well-defined and proving ...
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Let topological space $X$ and $f:X \to C$. Show that the space $X_{f}=\{g \in C(X) \mid \sup|g-f|<\infty\}$ is a complete metric space. [duplicate]

Let topological space $X$ and $f:X \to C$. Show that the space $$X_{f}=\{g \in C(X) \mid \sup|g-f|<\infty\}$$ is a complete metric space with distance function $$d(g_1,g_2)=\sup_{x \in ...
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Why in normed vector spaces we can define infinite series but in metric space we can not?

We usually define infinite series by partial sums and an inifinite series is said to converge if its partial sum converges. So, if $X$ is a normed vector spaces and $s_n=x_1+...+x_m$ is a partial sum ...
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1answer
16 views

Checking for varification: Proof - continuous function on a compact metric spaces is uniformly continuous

As a prep for my exam, I tried to prove all theorems we had during lecture on my own. I tried to prove that every continuous function on a compact metric space is also uniformly continuous. However in ...
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55 views

Notation question: Superscript on brackets around a set with a metric

Simplifying a bit, I encountered this notation: Where $\cal M$ and $\cal N$ are sets, each with a metric, $[{\cal M}]^{4\epsilon} \supset [{\cal N}]^{3\epsilon}$ . Any guesses as to the meaning of ...
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92 views

What is a “pseudonorm”?

The following is an excerpt of a note in topological vector spaces. I have tried to search "semi-pseudonorm" on Google but I have got nothing so far. A search with "pseudonorm" returns what we ...
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37 views

Show compactness

Let $X$ be a metric space, $K\subseteq X$ be compact and $C\subseteq X$ be closed. Use the definition to show that $K\cap C$ is compact. I am unsure as to which "definition" the question refers to. I ...
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Is the initial topology pseudometrizable?

For $i \in I$ and $k \in \mathbb{R}$, let $(X, \rho_i)$ be $k$-bounded pseudometric spaces. Let $\rho(x,x') = \sup_{i \in I} \rho_i(x,x')$ be the supremum pseudometric over $X$. It is true that ...
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57 views

Show that if $h \in C^1([a,b]\times[-ε,ε])$ with $ε>0$, then the function $s \in [-ε,ε] \mapsto \int_a^b h(t,s) \,dt$ is $C^1$

Show that if $h \in C^1([a,b]\times[-ε,ε])$ with $ε>0$, then the function $$s \in [-ε,ε] \mapsto \int_a^b h(t,s) \,dt$$ is $C^1$ and $${d\over ds}\int_a^b h(t,s) \,dt=\int_a^b {\partial h \over ...
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25 views

A metric in $\mathbb{C}^{\infty}$

Let $$d(z,z')=\frac{2|z-z'|}{\sqrt{1+|z|^2} \sqrt{1+|z'|^2}}, \mbox{if z,z' $\in \mathbb{C}$}$$ and $$d(z,z')=\frac{2}{\sqrt{1+|z|^2}}, \mbox{if z $\in \mathbb{C}$ and $z'=\infty$}$$ where $|z-z'|$ ...
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1answer
17 views

Is Jacobi weight power type or general type Sobolev weight?

Motivated by Shuhao Cao's answer in Weighted Poincare Inequality, I checked out Kufner's book weighted Sobolev spaces. Question Is the Jacobi weight either a power-type weight or a general weight in ...
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1answer
26 views

Metrics and Comparative Distances

Suppose $d_1$ and $d_2$ are metrics on a set M. I'm trying to find some insight to a condition where for all $w, x, y, z \in M$ then $d_1(w, x) > d_1(y, z) \iff d_2(w, x) > d_2(y, z)$, in other ...
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1answer
30 views

In a metric space $(X,d)$, for every Cauchy Sequence in $X$, and $z \in X$, the numeric succession $\{d(x_n;z)\}$ converges

Good night, I'm studying for a test and out teacher gives us a guide for it, but after trying i can't solve the last exercise, so any help would be appreciated. Prove that, for every Cauchy ...
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50 views

Show that $f(x)=\dfrac{d(x,F)}{d(x,F)+d(x,G) }$ is continuous.

In a metric space $M$, let $F=B[a,r]$ and $G=M-B(a,s)$, with $0<r<s$. Show that $f:M\to [0,1]$, define by $$f(x)=\dfrac{d(x,F)}{d(x,F)+d(x,G)}$$ is continuous and $f^{-1}(0)=F$ and ...
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14 views

Extending Rademacher's theorem

Rademacher's theorem states that Lipschitz functions are partially derivable of order $1$ almost everywhere. Are there conditions to be imposed on a continuous function $f : \Bbb R^n \to \Bbb C$ ...
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59 views

triangle inequality of metric d(x,y) [duplicate]

i need help on this problem: how can i prove that the metric d(x,y)=|x-y|/(1+|x-y|) from (ℝ,d) satisfies the triangle inequality? Greetings
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Is a metrizable space $X$ compact $\iff \exists x,y$ such that $d(x,y) = \sup d(u,v)_{u,v \in X}$?

Is a metrizable space $X$ compact $\iff \exists x,y$ such that $d(x,y) = \sup d(u,v)_{u,v \in X}$? Is the condition that $x,y$ at least necessary for the space to be compact? I have no idea how to ...
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For a Compact Metric Space $T: X\to X$ has unique fixed point

given: $(X,d)$ is a compact metric space $T:X\to X$ is such that $d(T(x),T(y))<d(x,y)\ \forall x,y\in X$ with $x\neq y$. Prove that T has a unique fixed point. Attempt: I think I can prove ...
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35 views

Show that mapping is uniformly continuous

Let $I=[0,1]$. Exhibit a subset of the metric space $C(I)$ (uniform metric), which is unbounded. Show that mapping $f\to \int_{0}^{1}f$ of $C(I)$ to $\mathbb{R}$ is uniformly continuous on $C(I)$ ...
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71 views

Establish isometries between $\mathcal{B}(X;M\times N)$ and $\mathcal{B}(X;M)\times \mathcal{B}(X;N)$

Establish isometries between $\mathcal{B}(X;M\times N)$ and $\mathcal{B}(X;M)\times \mathcal{B}(X;N)$, where $X$ is an arbitrary set, $M,N$ are metric spaces and $\mathcal{B}(X,M)$ represented the ...
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2 metric spaces have same convergent sequences, but only 1 is complete

Let $X=(0,1]$ and $$d_1(x,y)=|x-y|, \ d_2(x,y)=|\frac{1}{x}-\frac{1}{y}|$$ be two metric on $X$. Show that $(X,d_1), (X,d_2)$ have the same convergent sequences, but $(X,d_2)$ is complete while ...
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310 views

Proving T is a contraction

Trying to sure up some holes in my proof for the following: Let $I=[0,1]$ and define $T:C(I)\rightarrow C(I)$ by $$(Tf)(x)=x+\int_{0}^{x}(x-t)f(t)dt \quad \bigl(x\in I, f\in C(I)\bigr).$$ Show that ...
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32 views

Prove that $f^{-1}(N\times N-\triangle)$ is a union of open ball in M.

Let $f:M\to N\times N$ continuous with $M,N$ metric spaces and $\triangle\subset N\times N$ a diagonal. Prove that $f^{-1}(N\times N-\Delta)$ is a union of open ball in M. If f is continuous, ...
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How to prove $n^{\frac{1}{q}}\|x\|_p \le n^{\frac{1}{p}} \|x\|_q$ if $1\le p\le q$?

Let $1\le p\le q$ and $x\in \mathbb{R}^n$. Show that $$ n^{\frac{1}{q}}\|x\|_p \le n^{\frac{1}{p}} \|x\|_q, $$ where $\|x\|_k = (\sum_{i}|x_i|^k)^\frac{1}{k}$.
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Is it a given that subsets of metric spaces inherit their notions of distance from it?

Suppose I have a metric space $A$, for which there is a notion of distance $d$. If I were to make a subset of that space, is it a given that the subset will also have its distance defined by $d$? Or ...
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Prove that the closed unit ball is closed directly.

I'm trying to prove the following theorem, and I'm not sure my proof holds. Theorem. Let $(X,d)$ be a metric space, $p\in X$, and $r >0$. Then $$E = \left\{x\in X\ |\ d(x,p)\leq r\right\}$$ is ...
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1answer
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If $V$ the set of values of adherences of $(x_{n})$ then $V=\displaystyle\bigcap_{k=1}^{\infty}{F_{k}}$

A point $a$ in a metric space $M$ is called a value of adherence of sequence $(x_{n})$ in $M$, when $a$ is a limit of the one subsequence of $(x_{n})$. Let $V$ the set of values of adherences of ...
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Convergence of sequence of real numbers and (sub)subsequence

Let $(x_n)_{n\in \mathbb{N}}$ be a sequence of real numbers. A well known result is: The sequence $(x_n)_{n\in \mathbb{N}}$ converges to $a\in \mathbb{R}$ if and only if every subsequence ...
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What is the completion of the metric space of continuous functions with quadratic metric?

What is the completion of the metric space of continuous functions with metric $$d(f,g)=\sqrt{\int\limits_a^b(f-g)^2}$$ $f,g$ are function from $[a,b]$ to $\mathbb{R}$
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Showing that two topologies give rise to the same topology

Suppose that for a metric space $(X,d)$ we have another metric $p$ forming the metric space $(X,p)$. Both of these will give rise to two topologies $\tau_{1}$ and $\tau_{2}$. I want to show that ...
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Understanding proof that $X$ is compact if it is a metric space in which every infinite subset has a limit point

Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is compact. Proof: $X$ has a countable base. It follows that every open cover of $X$ has a countable ...
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Show that the product topology on $X \times Y$ is the same as the metric topology, where the metric is the product metric

Any help on the following problem would be greatly appreciated. Thanks! Given metrics $d$ and $e$ on sets $X$ and $Y$, let $f$ be the product metric on $X \times Y$. So ...
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128 views

Prove the map sending a matrix to it's inverse is continuous.

Let $GL(n,\mathbb{R})$ be the space of invertible $n \times n$ matrices over $\mathbb{R}$. I have already proved that $GL(n,\mathbb{R})$ is open in $\mathbb{R}^{n^2}$ and that it is not connected with ...
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481 views

Give examples of clopen (open and closed) sets

I was wondering whether somebody could help with the following problem. I think I have an answer, but I'm still a bit unsure so any help would be greatly appreciated. (a) Give an example of a ...
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Complete metric on the space of sequences

Let $S$ be the set of all real sequences $x=\{x_n\}$, $d: S\times S \rightarrow \mathbb R$ be defined by: $$d(x,y)=\sum_{n=1}^{\infty} \frac{|x_n-y_n|}{2^{n}[1+|x_n-y_n|]}.$$ Show that $(S,d)$ is a ...
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Determine if subset of $C[a,b]$ is Compact

Let $X = C[a,b] $ with the standard uniform metric. Let $g \in C[a,b]$ Prove that $F = \{f \in C[a,b] : |f(x)| \leq |g(x)| \; \; \forall x \in [a,b] \}$ is a compact subset $\iff g = 0$ $\impliedby$ ...
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Prove the general linear group is open in $\mathbb{R}^{n^2}$

I was wondering if somebody could help me with the following problem. Let $GL(n,\mathbb{R})$ be the space of $n \times n$ matrices over $\mathbb{R}$. Prove that $GL(n,\mathbb{R})$ identified in the ...
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35 views

Is it always possible to find a vector perpendicular to two given vectors in a general inner product space?

In short, given an inner product space $X$ , for any $x,y \in X$ does there always exist a nontrivial $z \in X$ so that $<x,z> = 0$ and $<y,z> = 0$ ?
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A metric space $(X,d)$ is compact iff every real valued continuous function on $X$ is bounded.

A metric space $(X,d)$ is compact iff every real valued continuous function on $X$ is bounded. I have got the solution to it which I dont get at all. Since in a metric space compactness is ...
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If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$

Let $f:[0,1]\to \mathbb{R}$ is continuous, such that $f(0)=f(1)$. For each $n\in\mathbb{N}$, prove that $\exists x\in [0,1]$ such that $x+\dfrac{1}{n}\in[0,1]$ and ...
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28 views

If $\partial X\subset Y$, then $X\cup Y$ is connnected.

Let $X,Y$ and $M$ connnected. If $\partial X\subset Y$, then $X\cup Y$ is connected. My approach: I know that the boundary of set $X$ is equal to $\partial X=\bar X\setminus int(X)=\bar ...
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Is the product of continuous functions in a metric space continuous?

I have the following problem: Let $(X,d)$ be a metric space, and let $f, g: X \rightarrow \mathbb{R}$ be continuous functions. Prove that $f + g$ (sum) and $fg$ (product) are continuous. I wrote the ...
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Counterexample with mapping

Let $f$ is mapping from metric space $X$ into metric space $Y$. Let $E$ some subset of $X$. Am I true if $f(x)\in f(E)$ then doesn't imply that $x\in E$? Let $f(x)=x^2$ and $X=Y=\mathbb{R}^1.$ Taking ...
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29 views

Open balls with $r > 1$ in real space with discrete metric

I've read a question Why is the open ball in a discrete space with radius 2 the metric space itself? where the answer to my question was given In a discrete metric space $M$, $d(x,y)=1$ ...
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36 views

An example of a metric space whose closed balls are never complete?

At first I thought that only the empty set could possibly fit the requirement. Since if there is any element in a metric space then the set containing even only one of the element, say $a$, would have ...
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177 views

What does it mean for a function to be bounded on its domain?

Suppose $S$ is an abstract metric space. Suppose $f(x):\Bbb{S} \to \Bbb{R}$. Then how what would the question be asking, if it asks "Show that $f$ is not bounded on $\Bbb{S}$"? Is it asking me to ...
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27 views

Can I say that $f(x)=0$ converges to zero?

we have a sequence $a_1$, with all terms larger than or equal to $0$. which has finitely many terms before the sequence becomes all $0$s. Now suppose I had a function = $min${of the values of $a_1$ ...
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If $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}^2$ are homeomorphic, then the interior of $Y$ is empty

Let $X\subset\mathbb{R}$ and $Y\subset\mathbb{R}^2$. If $X$ and $Y$ are homeomorphic, then $int(Y)=\emptyset$. My approach: If $X$ and $Y$ are homeomorphic, then exist a function $f$, such that ...
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Functions from $\mathbb{R}^{n}$ to $[0,1]$.

There's a question on which I am really stuck, but probably the answer might be very easy. Here is the statement. Consider ($\mathbb{R}^{n},d)$ [where $d$ is the Euclidean metric]. Let $x_{1}$ and ...
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1answer
52 views

Find closure of $G=\bigcup_{x\neq0}G_x$ [closed]

Let $G_x=\{(x, \cos\frac{1}{x})\}$ if $x\neq0$ and $G_0=\{(0,y):\, |y|\leq1\}$ in $\mathbb{R}^2$ with Euclidean metric. a) Prove that every point of $G_0$ is limit of a sequence in ...