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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When talking about a normed vector space, does it's metric always need to be the induced one?

The title basically says it all. If we have a normed vector space, is it possible to work with the space as a metric space with a different metric than the induced one? So if the space is $(X,||\ ...
5
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1answer
40 views

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...
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1answer
27 views

The closed and bounded sets are compact in the product topology

Let $X=\mathbb{R}^{\aleph_0}$ with the product topology, it is true that all the closed and bounded (in the uniform sense) sets are compact?
4
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1answer
41 views

Aggregating Metrics to Form a New Metric

I'm looking for a source or hints which could help me solve the following problem: Let $S$ be a set and let $d_i : S \times S \rightarrow [0,1]$ be a family of metrics for $i \in \{1, \ldots n\}$. ...
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0answers
10 views

Positive definite functions defined on the embedding of a planar graph in the plane

By way of motivation: Let $f:[0,1]\rightarrow\mathbb{R}^2$ and $g:\mathbb{S}_1\rightarrow\mathbb{R}^2$ be continuous injections, where $\mathbb{S}_1$ is the circle ($\mathbb{R}/\mathbb{Z}$). Then, ...
4
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2answers
69 views

Characterisation of Cantor-connectedness

For Cantor-connectedness I use the following definition: A $p$-metric space $(X,d)$ is Cantor-connected if for any $\epsilon > 0$, any two points $x, y \in X$ can be connected by an ...
2
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3answers
35 views

Show that the sequence of functions $(x_n)_{n≥1}$ in $C[0, 1]$ given by $x_n(t) = t^{2n} − t^{3n} , ∀t ∈ [0, 1]$ is bounded

That is $C[0,1]$ equipped with the supremum metric. I have proven, using derivatives, that each function $x_n$ has a local maximum and local minimum at $(2/3)^{1/n}$ and $0$ respectively. I know ...
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1answer
360 views

Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?

Let $X$ be a metric space. Suppose that the product $X\times\mathbb R$ admits a bi-Lipschitz embedding into $\mathbb R^{n}$. Does it follow that $X$ admits a bi-Lipschitz embedding into $\mathbb ...
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0answers
32 views

Problem 30 in the Exercises following Chapter 2 in Baby Rudin: How to immitate the proof of Theorem 2.43?

Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following ...
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1answer
26 views

Contraction Mapping, Metric

Let $X$ = {all continuous functions $f$:[0,1] $\rightarrow$ [0,1]} and let $d$ be the metric on $X$ given by $d$($f$,$g$)= $sup_{t\in[0,1]}$ |$f$($t$)-$g$($t$)| for $f$,$g$ $\in$ $X$ Show that ...
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2answers
58 views

Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?

Letting $\mathbb{Q}$ be equipped with the Euclidean metric. What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $. Its not ...
3
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1answer
49 views

Erwine Kryszeg's _Introductory Functional Analysis With Applications_: Section 2.3, Prob. 14

Here's problem 14 in the Problem Set immediately following Section 2.3 in the book, Introductory Functional Analysis With Applications by Erwine Kryszeg. Let $Y$ be a closed subspace of a normed ...
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3answers
43 views

Contraction-like mapping without fixed point?

If $(X,d)$ is a complete metric space and $\xi:\;X\to X$ satisfies: $$d(x,y)<n+1\Rightarrow d(\xi(x),\xi(y))<n$$ $$d(x,y)<1/n\Rightarrow d(\xi(x),\xi(y))<1/(n+1)$$ for all $n= 1,2,\dots$, ...
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2answers
163 views

A natural-looking distance formula

The distance formula in one dimension is $$D_1 = |x_2-x_1|$$ From the Pythagorean theorem, the distance formula in two dimensions is $$D_2 = \sqrt{|x_2-x_1|^2 + |y_2-y_1|^2}$$ Now, in three ...
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2answers
49 views
+50

Does this strengthening of continuity have a characterization in terms of familiar concepts?

Definition 0. Whenever $X$ is a metric space, $A \subseteq X$ is a subset, and $r \in \mathbb{R}_{>0}$ is a positive real number, define that $$A \oplus r = \bigcup_{a \in A} B_r(a).$$ Definition ...
3
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1answer
46 views

Adjoint of completely continuous operator is completely continuous

In the proof of the fact that the adjoint operator $A^\ast$ of a completely continuous linear operator $A:E\to E$ mapping a Banach space into itself is also completely continuous on $E^\ast$ endowed ...
2
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3answers
490 views

Analogue to Fixed Point Theorem for Compact metric spaces

If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
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1answer
33 views

Closedness and boundedness in metrizable topological spaces

This is a quick question that I have not managed to answer myself: let $X$ be a metrizable topological space, let $A\subset X$ be a closed, bounded subset. $f:X\to Y$ is a homeomorphism, must $f(A)$ ...
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0answers
24 views

Complete subspace of continuous function from compact subset [on hold]

Assume $K\in \mathbb{R}$ compact. How to prove that $C^0(K,\mathbb{R})$ is complete. Where $C^0(\mathbb{R},\mathbb{R})$ is the space of continuous f from $\mathbb{R}$.
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1answer
31 views

Example of metric space completion

I'm looking for examples of noncomplete metric spaces and their completions. I know of some basic examples like completion of open intervals and rational numbers(both with the reals and p-adic ...
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0answers
23 views

Structure of the space $X:=\{x \subset \mathbb{R}^3: |x| < \infty\}$, where $|x|$ is the cardinality of $x$

I am interested in the space $$ X:=\{x \subset \mathbb{R}^3: |x| < \infty\}, $$ where $|x|$ is the cardinality of the subset $x$. This is basically configuration space for a quantum system with a ...
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1answer
31 views

Topological structure of the Manifold valued functions

$M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is ...
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4answers
57 views

Show the subset $A$ of $\mathbb{R}^n$ is compact

Show the subset $$A = \{(x_1, . . . , x_n) ∈ \mathbb{R}^n| −1 ≤ x_1 ≤ x_2 ≤ · · · ≤ x_n ≤ 1\} \subset \mathbb{R}^n $$ is compact, and show the function $$\left\{\begin{array}{}f : A → ...
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1answer
267 views

proving definitions of connectedness are equivalent

Prove these definitions are equivalent: Definition $\,(1)\,$: $A\subset X$ is not connected if for open $U, V\subset X\,\,, $$\,\,U\cap\bar{V}=\emptyset\,\,$, $\,\,\bar{U}\cap V=\emptyset\,\,$, ...
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0answers
14 views

Extension theorem for locally Lipschitz functions

Let A be a subset of a metric space $(X,d)$ and f be a real valued locally lipschitz function on A. Does there exists a real valued locally lipschitz function on X which is an extension of f? or under ...
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1answer
45 views

Irrational numbers in [0,1] [on hold]

Why iirational numbers in interval [0,1] can't be countable union of closed sets?
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1answer
63 views

Integral equation and metric spaces

Let $C([0,\frac{\pi }{2}])$ be the set off all continuous functions defined on $[0,\frac{\pi }{2}]$ . Prove that this integral equation $$ f(t) = \int\limits_0^{\frac{\pi }{2}} {\arctan } ...
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1answer
30 views

How to prove a map between two spaces of real sequences $f : l^1 \to l^2 $ is well-defined and continuous

the question is whether the following statement is ture or false, and justify it. Here is the statement The map $f : l^1\to l^2$ given by $f(x_0, x_1, x_2,...)= (x_0, x_1, x_2,...) $ is ...
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0answers
7 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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1answer
18 views

not metrizable?

In Munkres, section 30, exercise 6 is this: Show that $R_{l}$ and $I^2_0$ are not metrizable. I guess $R_{l}$ is lower limit topology, and $I^2_0$ is an ordered square. and here, how to prove they ...
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2answers
568 views

Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty

The following is problem 3.22 from Rudin's Princples of Mathematical Analysis: Suppose $X$ is a nonempty complete metric space, and $\{G_n\}$ is a sequence of dense open subsets of $X$. Prove ...
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1answer
16 views

How to find the interior of this set?

let $S=\{A\in M_n(\mathbb R):tr(A)=0\}$ The question is to check whether $S$ is Nowhere dense .I think the set is closed and hence the problem reduces to findind int(S).How to do that?
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1answer
29 views

$\{(x,y)\in \mathbb R^2:xy=1\}$

To check which pairs are Homeomorphic? A.$\{(x,y)\in \mathbb R^2:xy=0\}$ B.$\{(x,y)\in \mathbb R^2:xy=1\}$ C.$\{(x,y)\in \mathbb R^2:xy=0,x+y\geq0\}$ D.$\{(x,y)\in \mathbb R^2:xy=1,x+y\geq 0\}$ I ...
1
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1answer
52 views

Approximation of $f\in L_p$ with simple function $f_n\in L_p$

Let us use the definition of Lebesgue integral on $X,\mu(X)<\infty$ as the limit$$\int_X fd\mu:=\lim_{n\to\infty}\int_Xf_nd\mu=\lim_{n\to\infty}\sum_{k=1}^\infty y_{n,k}\mu(A_{n,k})$$where ...
1
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2answers
26 views

Is the sequnce of functions $f_n$ convergent?

let $f_n$ be a sequence of functions in $C[0,1]$ and they are differentiable continuously in $(0,1)$. Also $|f_n(x)|\leq 1$ and $|f_n{'}(x)|\leq 1$ forall $x\in [0,1]$ and for each $n$. Since $[0,1]$ ...
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1answer
22 views

How to check the compactness of these sets:

How to check the compactness of these sets: a.the unit sphere in $l_2$ the space of all square summable real sequences with its usual metric i.e.$d({x_i,y_i}) =(\sum_1^\infty|x_i-y_i|^2)^{1/2}$ ...
1
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2answers
38 views

Continuity of vector space operations in a normed space

Here's problem 4 immediately following section 2.3 in Erwine Kryszeg's book, Introductory Functional Analysis With Applications: Show that in a normed space $X$, vector addition and scalar ...
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0answers
26 views

Metrizable topological space

Why Extended real numbers set with T ( T topology on R with infinity) , is metrizable ? And how can prove that d(x,y) genetares this topology (T) ??
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1answer
81 views

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$? My goal is to claim that in any finite dimensional vector space, equipped with a metric, a closed-bounded subset ...
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2answers
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Sequences in Metric Spaces [closed]

Find a metric space $(X,d)$ and a sequence $(x_n)$ in X that has no convergent subsequences but for which the infimum of the set $\{d(x_m,x_m)\mid m$ and $n$ are distinct natural numbers$\}$ is zero. ...
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2answers
38 views

Canonical compactification of a metric space

There are many constructions to produce a compact metric space from an arbitrary metric space (sometimes extra conditions are imposed). But is it possible to compactify a metric space M into M* such ...
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1answer
13 views

Multiple choice question on continuous function on a unit ball

Pick out true: Let $B$ be the closed unit ball and $D$ be the open unit ball. a.Given a continuous function $g:B\rightarrow \mathbb R$ there always exists a continuous function $f:\mathbb ...
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4answers
287 views

Is it true that any metric on a finite set is the discrete metric?

Is it true that any metric on a finite set is the discrete metric? I can see that it's at least equivalent with the discrete metric since $B(x,\delta)=\{x\}$ where $X=\{a_i\}_{i=1}^n,$ ...
0
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0answers
27 views

Showing that $\mathcal{H}^s$ is Borel regular (assuming we know already know that $\mathcal{H}^s$ is measure)

I am trying to show that $\mathcal{H}^s$ (s-dimensional Hausdorff measure) is Borel regular. I am using the defintion $\mathcal{H}^s_{\delta}(F)=inf \Bigg\{ \sum_{i=1}^{\infty}|V_i|^s : \{V_i\} \text{ ...
1
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0answers
27 views

Prove this map is continuous

$(rcos(t),rsin(t))↦((1/r).cos(t),(1/r).sin(t)), 0≤t≤2pi $ first for $0<r<1$, then for $r>1$ My idea is to say $(rcos(t),rsin(t)) = r .(cos(t),sin(t))$ then the cos and sin map with an ...
1
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0answers
52 views

Calculating Euclidean dissimilarity for a given cluster with itself

Suppose I have clusters $$A= \{(1,1)^T, (1,2)^T\} $$ $$B=\{(2,3)^T, (3,4)^T\} $$ $$C= \{(4,5)^T, (5,6)^T, (1,2)^T\} $$ I wish to use the Euclidean dissimilarity and Average linkage to calculate a ...
1
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1answer
24 views

Show that an irregular 1-set in the plane is totally disconnected

A $1$-set is a Borel set such that $0 < \mathcal{H}^1(A) < \infty$, where $\mathcal{H}^s$ is the Hausdorff measure. Let $A$ be an irregular $1$-set in the plane. Deduce from the theorem below ...
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2answers
24 views

Check if the parabola (with an induced topology) $\{(x,y)\in\mathbb{R}^2 | y=x^2\}$ is connected or compact.

i think yes connected but not compact, as it cannot be represented as a disjoint union and there is no finite sub cover. I'm just not sure how to go about proving this i.e. what to actually write ...
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votes
1answer
56 views

Show these sets are homeomorphic to eachother

1) {${(x, y) ∈ R^2 |0 < x^2 + y^2 < 1}$} 2) {${(x, y) ∈ R^2 | x^2 + y^2 > 1}$} I've considered mapping r to 1/r, from (0,1) to (1,∞)
0
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2answers
24 views

Prove $d(x,y)=\sup _{n} \left| \sum _{k=1}^{n}(x_k-y_k)\right |$ is a metric

Let $\gamma$ be the set of convergent series.$$\gamma = \{x=(x_k), x_k \in \mathbb{R} : \sum x_k <\infty\}$$ Prove that $(\gamma , d)$ is a metric space, with $$d(x,y)=\sup _{n} \left| \sum ...