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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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, ...
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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 = ...
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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 $d_2$, ...
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31 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
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35 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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} - ...
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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 ...
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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 $d$?
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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 $\{x_0, x_1, \ldots, x_n\}$ ...
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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 ...
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48 views

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

$$d_2, d_1-\text{metrics in } R^k$$ $$d_2(x,y)=(\sum_{i=1}^{k}|x^i-y^i|^2)^{1 \over 2} \\ d_1(x,y)=\sum_{i=1}^{k}|x^i-y^i| \\ d_2(x,y) \leq d_1(x,y) \leq \sqrt{k}\ d_2(x,y)$$
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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.) ...
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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 ...
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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
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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 ...
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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 ...
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1answer
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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. ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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72 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 ...
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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 ...
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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 ...
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1answer
14 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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) + ...
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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 ...
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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)?
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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 ...
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1answer
61 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 ...
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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
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0answers
51 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: $$B_{(\mathbb{R}^2,d_{l_2})}((0,0),1) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 <1\}$$ ...
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0answers
33 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?
2
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1answer
51 views

Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space) "Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed ...
7
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1answer
100 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
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1answer
59 views

Why is $f(x,y)$ said to be discontinuous at $(0,0)$?

Why is $f(x,y)=\begin{cases} \frac{x^2y}{x^4+y^2}, & \text{if $(x,y)\neq (0,0)$}\\[2ex] 0, & \text{if $(x,y)=(0,0)$} \end{cases}$ said to be discontinuous at $(0,0)$? I am supposed to show ...
2
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3answers
36 views

If $K\in \Bbb{R}^n$ is compact, $\sup_{x,y\in K}|x-y|=\max_{x,y\in K}|x-y|$.

Suppose $K\in \Bbb{R}^n$ is compact. Let us denote $D=\sup_{x,y\in K}|x-y|$ as $K's$ diameter. Prove there exist $a,b\in K$ such that $D=|a-b|$ i.e, that the suprimum is the maximum. I know there ...
0
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1answer
45 views

Show that the space $ℓ^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0\text{ for } j>>1\}$ is not complete

Show that the space $$\ell^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0 \text{ for } j\gg1\}$$ with inner product $$(a,b) \in ℓ^0\timesℓ^0 \mapsto \langle a,b\rangle =\sum_{j=1}^\infty ...