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.

learn more… | top users | synonyms (1)

0
votes
1answer
105 views

Length of unit circle

Let $\it{l} $ be the length of the unit circumference $\{(x,y):||(x,y)||=1\}$ in an arbitrary norm $||\cdot||$ in $\mathbb{R}^2.$ How to prove or disprove the inequalities $\it{l} \ge 6,\, \it{l} \le ...
1
vote
1answer
15 views

Suppose $X$ is a metric space and $S \subseteq X.$ Then, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.

Suppose $X$ is a metric space and $S \subseteq X.$ Then, according to my textbook, $S^o=\{x \in X~|~dist(x,S^c)>0\}$. (Notations Used: $S^o$ refers to interior of $S$ . If $x \in X, dist(x,S) = ...
1
vote
4answers
45 views

A subset $U$ of a complete metric space such that all continuous functions on $U$ attain a minimum must be closed

I was working on the following problem: Suppose we have a complete metric space $(X,d)$. Show that if every continuous function on a subset $U \subset X$ attains a minimum, then $U$ is closed. ...
2
votes
2answers
58 views

The topological space $\left(X,2^X\right)$ is metrizable

Prove that for each set $X$, the topological space $\left(X,2^X\right)$ is metrizable, where $2^X$ is the power set. What I'm not sure is what are the conditions for a topological space to be ...
3
votes
2answers
61 views

Prove that $U-f(U)$ is an open set.

Let $(X,d)$ be a compact metric space. Let $f:X\to X$ be continuous. Fix a point $x_0\in X$, and assume that $d(f(x),x_0)\geq 1$ whenever $x\in X$ is such that $d(x,x_0)=1$. Prove that $U\setminus ...
2
votes
1answer
40 views

A possible complete metric on the set of twice-differentiable functions on $[0,1]$

Let $S \subset C^2[0,1]$ (set of two times differentiable functions $f(x)$ on $[0,1]$) which satisfy the following: $$\int_0^1 f(x)\,dx\leq3$$ Question is $(S,d)$ is a complete metric space, ...
1
vote
2answers
45 views

Prove that the normed spaces $(C[0,1], \| \cdot\|_2)$ and $(C[a,b], \|\cdot \|_2)$ where $\| \cdot\|_2$ is the Euclidean norm are isometric.

Essentially I'm looking for a bijection $f: C[0,1]\to C[a,b]$ such that$$\|f(x) \|_2=\| x\|_2$$ I don't know how to go about finding this function, but I do know that it is possible. $$\| x \|_2 = ...
0
votes
2answers
67 views

If $d_1(x,y)$ and $d_2(x,y)$ are metrics, prove that $d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$ is a metric.

$$d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$$ The first three properties are trivially proven. The triangle inequality, not so much. I tried using the triangle inequalities that apply to $d_1$ and $d_2$, ...
1
vote
1answer
37 views

Define a metric using scalar product and prove that it is indeed a metric

So this is how I went about this: $\langle\,\cdot\,,\,\cdot\,\rangle: X \times X \to \mathbb{R}$ such that (by definition I list the properties of scalar product) and I can east prove the first three ...
2
votes
1answer
37 views

Graph of a non-continuous function is closed

Exercise. Let $\ f\colon\mathbf R\to\mathbf R$ be defined by $$f(x)=\begin{cases}\frac{1}{x},\ x>0\\0,\ x\leq 0\end{cases}$$ Prove that the graph $\Gamma_f:=\{(a,f(a)):a\in\mathbf R\}$ of ...
2
votes
1answer
31 views

Proof that every polilinear map who's domain is $R^{n_1} \times R^{n_2}… \times R^{n_k}$ and co-domain any given real normed space Y is bound.

A Polilinear map\operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
1
vote
1answer
42 views

Can anyone help out with this proof, certain steps are unclear. Norm of linear operator.

I have the following norm defined as follows (in $R^n$, $x=(x^1,x^2,\ldots,x^n)\ $)$\| x\|_1= \sum_{i=1}^{n}|x^i|$ Let $A:R^m \to R^n$ a linear map of the spaces $(R^m ,\| \cdot \|_1 )$ and $((R^n ...
3
votes
1answer
77 views

Let $A ,B \subseteq \Bbb{R}^{k}$ and $A+B =\{a+b \mid a\in A, b\in B\}$then:

Let $A ,B \subseteq \Bbb{R}^k$ and $A+B =\{a+b \mid a\in A, b\in B\}$then: a)If $A,B$ be open then $A+B =\{a+b \mid a\in A, b\in B\}$ is open. b)If $A,B$ be connect then $A+B$ is connect? c)If ...
2
votes
1answer
37 views

Proving that $f \big|_{\partial A} = 0$, where $A = [f> 0]$.

I found this exercise: let $M$ be a metric space, and $f: M \to \Bbb R$ be a function, and $A = \{ x \in M \mid f(x) > 0\}$. Prove that if $x \in \partial A$, then $f(x) = 0$. I think that we must ...
19
votes
1answer
254 views

Let $f:K\to K$ with $\|f(x)-f(y)\|\geq ||x-y||$ for all $x,y$. Show that equality holds and that $f$ is surjective. [duplicate]

$K$ is a compact subset of $\Bbb R^n$ and $f:K\rightarrow K $ satisfies : $$\|f(x)-f(y)\|\geq \|x-y\|$$ Show that $f$ is bijective, and that : $$\|f(x)-f(y)\| = \|x-y\| $$ It's easy to show that ...
0
votes
4answers
38 views

If $d(x,y)$ is a metric, how does the following inequality apply?

I'm interested if someone can formally type out why this is. I thought it was trivial, but the professor wanted a more detailed explanation: $${d(x,y)\over {1+d(x,y)}}\leq ...
2
votes
3answers
39 views

Can anyone prove the second property of a the following metric? $d: C([a,b])\times C([a,b]) \to R ; \ \ \ d(f,g)=\int_{a}^{b}|f(t)-g(t)|dt$ [closed]

$$d: C([a,b])\times C([a,b]) \to R ; \ \ \ d(f,g)=\int_{a}^{b}|f(t)-g(t)|dt$$ $2.) d(f,g)=0 \iff f \equiv g$ Now in my notebook some lemma is called upon, concerning integrals, but it is unclearly ...
2
votes
2answers
29 views

If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics.

If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics defined. Is the diameter of a subset unique with respect to the two most ...
1
vote
1answer
21 views

Can varying $p$ in the $p$-norm induced distanced change which pair of points are closer?

i.e. For some $x, y, z \in \mathbb{R}^{n}$, do there exist $p_{1}, p_{2}$ s.t. $ 0 < \sum_{i=1}^{n} (|x_{i} - y_{i}|^{p_{1}} - |x_{i} - z_{i}|^{p_{1}})$ and $ 0 \ge \sum_{i=1}^{n} (|x_{i} - ...
0
votes
1answer
29 views

Finite closed covering of a bounded set in $\mathbb{R}^n$

My Attempt: I think here I can define the diameter of $A$ as follows since it is bounded. diam $A=\sup \{|x-y|: x, y \in \mathbb{R}^n\}$ So, I can take each $r_k$ as diam $A$. Am I on the ...
-1
votes
1answer
34 views

A new metric formed from the old [duplicate]

Let $(X,\rho)$ be a metric space. Is it true that $d:=\dfrac{\rho}{1+\rho}$ is also metric. If it's true can anybody give a hint how to prove the triangle inequality for $d$?
0
votes
1answer
61 views

compact image of a continuous function from compact set to C

Suppose that we have a continuous function $h:[0,1] \times [a,b] \to G$, where $G$ is an open subset of $\mathbb C$. Prove that we can partition $[0,1]$ and $[a,b]$ to $\{x_0, x_1, \ldots, x_n\}$ ...
0
votes
1answer
41 views

A metric on the set of closed bounded subsets of a metric space

Define the distance from a point $p$ in a metric space $(X,d)$ to a subset $Y \subset X$ by $$d(p,Y) := \inf \{ d(p,y) : y \in Y \}.$$ For any $\varepsilon > 0$, define $$Y_\varepsilon := \{ x ...
0
votes
1answer
52 views

Why does this inequality hold, formally looking at it? Can someone prove this?

$$d_2, d_1-\text{metrics in } R^k$$ $$d_2(x,y)=(\sum_{i=1}^{k}|x^i-y^i|^2)^{1 \over 2} \\ d_1(x,y)=\sum_{i=1}^{k}|x^i-y^i| \\ d_2(x,y) \leq d_1(x,y) \leq \sqrt{k}\ d_2(x,y)$$
1
vote
2answers
27 views

Question on metric spaces.. 2 properties which I don't know whether they apply

Do these two properties hold in all metric spaces. In my textbook, it says they hold in spaces, that have defined scalar products, but I am interested if they hold in generally metric spaces: $$1.) ...
2
votes
2answers
37 views

Prob. 2 (e), Sec. 27 in Munkres' TOPOLOGY, 2nd ed: Open supersets and $\epsilon$-neighborhoods of closed noncompact sets

This question concerns exercise 2(e) from section 27 (p.177) in Munkres' Topology: Let $(X,d)$ be a metric space, and let $A$ be a non-empty subset of $X$. For any point $x \in X$, we ...
2
votes
2answers
90 views

Compact set in a open set

Why the distance of compact set from the boundary of open set it is contained in has to be greater than 0. i.e. G open set and S is compact set contained in G, prove dist(S, G) >0
0
votes
2answers
36 views

Is it true that a mapping between metric spaces is continuous iff the image of every open set is open?

Just want to change Rudin theorem 4.8 a bit and see if this works. The original theorem is ... $f$ is continuous iff $f^{-1}(V) $ is open in $X$ for every open set $V$ in $Y$. If I change the ...
1
vote
4answers
42 views

Closed set in $l_{2}$

I need to show that the set $A=\left\{ x \in l_{2} : |x_{n}| \leq \frac{1}{n}, n=1, 2, ...\right\}$ is a closed subset of $l_{2}$ I'm assuming the best way to show this is to have a sequence in A ...
1
vote
1answer
45 views

Equivalence of “sequence that admits a cauchy subsequence”

Let $S$ be a subset of a metric space $(X,d)$. I have read (here) the "Sequential characterization of totally bounded subsets" that says the following are equivalent: 1.) $S$ is totally bounded. ...
5
votes
2answers
57 views

Open set in a general metric space.

Let d define a metric on an infinite set $M$. Show that there exists an open set $U$ such that $U$ and its complement are infinite. (Infinite referring to cardinality in both instances) I know this ...
0
votes
1answer
19 views

For every $n \ge 2 $ , existence of uncountably many mutually disjoint closed balls , complement of whose union is path connected

For every $n \ge 2 $ , does there exist an uncountable family $\{D_\lambda: \lambda \in \Gamma\}$ of mutually disjoint closed balls in $\mathbb R^n$ , such that $\mathbb R^n \setminus \cup_{\lambda ...
3
votes
2answers
42 views

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ?

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ? Or , does every infinite dimensional normed linear space has ...
0
votes
4answers
51 views

For a finite set in $\mathbb{R}$, the interior is empty and the closure and boundary are the set itself

How do I show explicitly that for a finite set in $\mathbb{R}$ the interior is empty and the closure and boundary are the set itself? For closure is simple: it is union of boundary and interior.But ...
0
votes
2answers
81 views

Open ball metric space vs open set topological space

I'm having trouble understanding the notion of an open set when applied to a space without a metric defined on it - I have read that all metric spaces are naturally a topological space, but the ...
7
votes
1answer
73 views

Topological space $\nRightarrow$ Metric space $\nRightarrow$ Normed space $\nRightarrow$ Inner product space (Examples)

If I have an inner product space, the hierarchy goes: Inner product space $\Rightarrow$ normed space $\Rightarrow$ metric space $\Rightarrow$ topological space. The reverse, however, is not always ...
3
votes
1answer
59 views

How to show that the space of polynomials is not complete

Denote by $P[0,1]$ the set of all polynomials $p\colon [0,1]\to\mathbb{R}$; this is a vector space. Endow $P[0,1]$ with the norm $$\| p\|=\sup_{t\in [0,1]}{| p(t)|}.$$ I want to show that this ...
0
votes
1answer
15 views

Upper semi-continuity results

I have recently been introduced to the notion of upper semi-continuity on a metric space $X$. Please advise on the following queries: If $f:X \rightarrow \mathbb{R}$ is upper semi-continuous and ...
1
vote
3answers
60 views

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 ...
18
votes
2answers
431 views

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 ...
1
vote
0answers
19 views

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 ...
1
vote
0answers
83 views

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 ...
1
vote
0answers
19 views

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) + ...
1
vote
0answers
32 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 ...
0
votes
1answer
29 views

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)?
-1
votes
1answer
52 views

Calculating distance between two squares of a board

Given an $n\times n$ board, for example a chess board 8x8, with the squares ordered in a Little-Endian Rank-File Mapping. Is there a direct way to calculate the distance between two squares using ...
1
vote
1answer
64 views

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 ...
0
votes
2answers
19 views

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{ ...
2
votes
0answers
52 views

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: $$B_{(\mathbb{R}^2,d_{l_2})}((0,0),1) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 <1\}$$ ...
2
votes
0answers
34 views

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?