Tagged Questions
0
votes
0answers
62 views
Metric on the space of Lipschitz continuous functions
Let $X=C^{[0,1]}([0,1])$, the set of Lipschitz continuous functions with domain $[0,1]$.
a. Prove that
$$\rho(f,g) := \sup|f-g|+\operatorname{Lip}(f-g)$$
is a metric on $X$.
Recall that
...
1
vote
3answers
71 views
A condition that balls have finite measure
Let $(X,d)$ be a metric space and let $\mu$ be a positive measure on $X$. I want to require that $(X,d)$ and $\mu$ have either of the following properties:
$\forall y \in X$, $\forall r \geq 0$, ...
1
vote
2answers
46 views
Necessary and Sufficient Condition for two metrics to have same open sets.
There are couple of independent conditions like one being scalar multiple of another, or if $$d_p(x,y)=(x^p+y^p)^{1/p}$$ then all $d_ps$ and $d_qs.$ which guarantee that open sets are same under these ...
1
vote
1answer
41 views
sequences of functions $\cos(\frac{x}{n})$
I have the sequence $(f_n)$ of functions $f_n: [-1, 1] \rightarrow \mathbb{R}$, defined by $f_n(x) = \cos\left(\frac{x}{n}\right)$. I need to show that $(f_n)$ is Cauchy in the space $(C[-1, 1], d)$ ...
1
vote
0answers
44 views
Determining Complete Metric Spaces
I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$
My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
0
votes
1answer
31 views
Correctness of Analysis argument with Cauchy sequences
Let $(x_n)$ and $(y_n)$ be Cauchy sequences in $(X, d)$. Show that $(x_n)$ and $(y_n)$ converge to the same limit iff $d(x_n, y_n) \rightarrow 0$
Proof $\rightarrow$
Suppose $(x_n) \to a$ and $(y_n) ...
1
vote
1answer
35 views
Cauchy Sequences and Analysis
Let $(x_n)$ and $(y_n)$ be Cauchy sequences in a metric space $(X, d)$. Show that the sequence $(d(x_n, y_n))$ is a cauchy sequence in $\mathbb{R}$.
What is the significance of $\mathbb{R}$ in this ...
3
votes
2answers
72 views
Defining a metric space
I'm studying for actuarial exams, but I always pick up mathematics books because I like to challenge myself and try to learn new branches. Recently I've bought Topology by D. Kahn and am finding it ...
3
votes
1answer
111 views
Unit ball of a Separable Banach Spaces is metrizable
Please help me to understand why the unit ball of a separable banach space is metrizable, when it is given the induced topology from the weak topology. Specifically, my problem is I think I'd be able ...
2
votes
2answers
114 views
Is this function Lipschitz continuous?
Let $\mu \in \mathbb R^d$ be given. Is the function $f:\mathbb R^d \to \mathbb R^d$ defined as $f(x) := \exp(-\|x- \mu\|) (\mu - x)$ Lipschitz continuous?
More specifically, for any $x, y \in ...
5
votes
3answers
75 views
Grasping the definition of open and closed sets
In a metric subspace $S = [0,1]$ of $\mathbb{R}^1$, why is it that every interval of the form $[0,x)$ or $(x,1], x\in (0,1)$, is an open set in $S$?
I understand that if you were to remove either, ...
3
votes
0answers
50 views
What are norms used for?
These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
3
votes
1answer
94 views
Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous?
Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$.
Is $f$ continuous?
Let $f$ be ...
2
votes
1answer
30 views
Maximum volume change for two sets with small Hausdorff metric in bounded part of $\mathbb{R}^n$
Given two subsets $S_1$, $S_2$ of a bounded part of $\mathbb{R}^n$, say $[-M,M]^n$. Is there a way to relate the difference in volume $vol(S_2)-vol(S_1)$ to the Hausdorff metric distance between the ...
1
vote
2answers
50 views
A question about metric spaces
Assume that we have a metric space $(S,d)$ and points $a,b,c \in S$ which statisfy the following conditions:
for all $x \in S$, $d(a,x) \leq d(a,b)$,
for all $y\in S$, $d(b,y) \leq d(b,c)$.
Does ...
2
votes
1answer
31 views
Correctness of Converging sequence and Adherent Points
$x\in X$ is an adherent point of $A\subset X$ if for every $\epsilon>0$ there exists $y\in A$ s.t. $y\in B(x, \epsilon)$
$B(x, \epsilon)$ is the open ball centered at $x$ with radius $\epsilon$
...
2
votes
1answer
138 views
For any point $ a $ of a compact subset $ S $ of a metric space, prove that there exists a nearest point $ c $ to $ a $.
Let $S$ be a compact subset of $X$. Define a metric space $(X, p).$ Prove that for any point $a\in X$, there exists a nearest point $c$ in $S$ to $a$. Moreover, $c$ in $S$ such that $p(c,a)\leq ...
0
votes
0answers
22 views
Equivalent metrics and inclusion of balls [duplicate]
I need some help with the following proof. I am stuck. My general idea is that if $d_1$ and $d_2$ are equivalent metrics then the balls converge to the same point? However, my understanding of metric ...
0
votes
1answer
53 views
Let (Y,d) be a complete metric space, and let G be a family of continuous functions from Y to $\mathbb R$…
Let $(X,d)$ be a complete metric space, and $F$ be a family of continuous functions from X to $\mathbb R$. Suppose that for each $x\in X$ there exists $M_x\in$$\mathbb R$ such that $f(x)\le M_x$ for ...
3
votes
2answers
52 views
Second Countability of Euclidean Spaces
Sorry I know this is a stupid question. However I got stuck on this for quite a while. I'm trying to prove that Euclidean spaces have a countable base, which can be constructed by taking all the open ...
2
votes
2answers
136 views
If $f_n(x)=x^n$ converges to $f$, why is $f$ not continuous?
I was reading my Analysis course notes and had some trouble. I hope you can help me.
Let $C(X)=\{ f | f:X \longrightarrow \mathbb{R} \text{ is a continuous function}\}$.
It was already stated and ...
2
votes
2answers
115 views
Metric Spaces Analysis
Let $(X,d)$ be a metric space and for $x,y \in X$ define
$d_b(x,y) =$ $ \dfrac{d(x,y)}{1 + d(x,y)}$
a) show that $d_b$ is a metric on $X$
Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$
...
1
vote
2answers
60 views
Closure in Metric Space
I need help understanding Theorem 2.27(c) in Rudin.
If $X$ is a metric space and $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\bar ...
1
vote
1answer
39 views
Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$.
Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$.
My workings so far: Because this is an if and only if proof we need to show it both ways. First let's assume ...
3
votes
1answer
100 views
Is $\mathbf{R}^\omega$ in the uniform topology connected?
Let $\mathbf{R}^\omega$ be the set of all (infinite) sequences of real numbers. Then is this space connected in the uniform topology? How to determine this?
The uniform metric $p \colon ...
0
votes
2answers
44 views
How to determine if this map is open or closed?
Given two supspaces $X:= [0,1]\cup[2,3]$ and $Y:=[0,2]$ of $\mathbf{R}$, let $f \colon X \to Y$ be defined as follows: $$f(x):= \left\{ \begin{array} {ll} x & \mbox{if $0\leq x\leq 1;$} \\ x-1 ...
1
vote
1answer
46 views
About Convergence of the Image of a Convergent Sequence Under a Uniformaly Convergent Sequence of Functions
Let $X$ be a topological space and $Y$ a metric space. Let $f_n \colon X \to Y$ be a sequence of continuous functions. Let $x_n$ be a sequence of points of $X$ converging to a point $x \in X$. Suppose ...
3
votes
1answer
39 views
Generalizations of derivatives using distance measures
Let $d(x,y)$ be a distance metric for two points $x,y\in \mathbb{R}^p$. Further, suppose that there are two real or complex sequences $X_n(x)$ and $X_n(y)$, $n=1,2,\ldots$ that depend on $x$ and $y$ ...
2
votes
2answers
158 views
Homeomorphic and Isometric Spaces
Problem
I'm currently studying metric spaces, and the lectruer's notes make the remark:
Clearly $(0,1)$, $(0,\infty)$ and $\mathbb{R}$ are homeomorphic under the standard metrics, but no two of them ...
1
vote
2answers
46 views
The set of points whose distance to a set $E$ in $\mathbb R^n $ is zero, is the same set $E$?
If a set $E$ is contained in $\mathbb R^n$ with the standard euclidean norm and if define another set $B$ as the points in $\mathbb R^n$ whose distance to the set $E$ is zero, is it true that $E=B$?
...
4
votes
1answer
159 views
Metrizability of weak convergence by the bounded Lipschitz metric
Why is the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ metrizable by the bounded Lipschitz metric $$d(\mu, \nu) = ...
2
votes
4answers
114 views
Addition of points on Metric Space
Well, I was not quite aware that addition of points is not defined in metric spaces but is defined only on linear spaces and others.
Could anyone elaborate why is this?
Is the addition of intervals ...
0
votes
2answers
117 views
Metric spaces question
\begin{equation}
P = \{f \in\ C ( \mathbb{R}, \mathbb{R}) \mid f(x+2 \pi ) = f(x)\}
\end{equation}
be the set of $2\pi$-periodic function.
1) Show that $P$ is a subspace of $C( \mathbb{R}, ...
0
votes
0answers
133 views
Isometries of metric spaces questions
A function $f$ from a metric space $(E, d)$ onto a metric space $(Y,\tilde d)$ is called an isometry if
$$ \forall x, y \in E : \tilde d(f(x),f(y)) = d(x,y). $$
Show that the function $f:(0,1] ...
4
votes
4answers
100 views
Show that the set given is closed
A question that I encountered which looks different than a normal open/closed sets proofs:
Let $(E, d)$ be a metric space, let $f : E\to R$ be continuous and $a$ element of $R$. Show that the set
...
3
votes
4answers
164 views
Show that the set is closed
Let $(E, d)$ be a metric space, $x$ element of $E$. Show that the set
\begin{equation}
A = \{y \in\ E : d(x, y) \geq 5 \}
\end{equation}
is closed.
Generally, how would you go about this?
I have an ...
0
votes
2answers
270 views
Boundary and Interior Points of the set: Rational Numbers
While I was studying metric spaces, I saw this question on the book:
Let $Q$ be the set of all rational numbers. What are $intQ$ and $∂Q$?
I think the answer should be: $intQ$ is all the ...
2
votes
2answers
89 views
Lipschitz property and Lipschitz extension
Is there a Lipschitz function $f$ from a subset of a metric space $U$ to a complete metric space $V$ that has no Lipschitz extension to the whole space $U$?
0
votes
1answer
54 views
Prove if one set is complete then another set is complete
Let $X$ be a set. Let $l^{\infty}(X,N)$ be all bounded functions on the form $f: X\longrightarrow N$. Let $d(f,g)=\sup\{n(f(x),g(x): x\in X)\}$ be a metric on $l^{\infty}(X,N)$, where $n$ is metric on ...
2
votes
3answers
172 views
Is $x^n$ Cauchy in $(C[0, 1], ||\cdot||_{\infty})$?
Consider the sequence of functions
\begin{equation}
f_n(x) = x^n, \quad x \in [0, 1].
\end{equation}
Is this sequence Cauchy in $(C[0, 1], ||\cdot||_{\infty})$?
The pointwise limit is not ...
2
votes
0answers
73 views
Prove metric space…
Let $l=\{(a_n)_{n \in ℕ}|a_n \in \Bbb R \text{ and } \sup|a_n|<\infty\}$
If $a=(a_n)$ and $b=(b_n)$ are in $l$, define a metric on $l$ by
$$d(a,b)=\sup_{n \in \Bbb N}|a_n-b_n|$$
Prove that $d$ is ...
0
votes
1answer
86 views
What is the intersection of this infinite amount of open sets?
Say I have the following
$$\bigcap_{n \in \mathbb{N}} \left(-\frac{1}{n}, \frac{1}{n}\right)$$
What is the value of this intersection of an infinite amount of open sets?
0
votes
2answers
75 views
show that $(-1,1)$ and $\mathbb{R}$ are isometric
I want to find a metric $d'$ sucht that $(\mathbb{R}, d')$ and $(X, d)$ with $X = (-1,1)$ and $d(x,y) = |x-y|$ are isometric. I tried the homeomorphisms
$$
f:\mathbb{R} \to (-1,1) ~ \textrm{ with } ~ ...
5
votes
0answers
134 views
Are these sets in $\mathbb{R}$ open and/or closed?
In $\mathbb{R}$, are these sets open? Are they closed?
$A = \{\frac{1}{n} : n \in \mathbb{N}\}$
$B = A \cup \{0\} $
$[0, 1)$
My thoughts:
$A$ is not open as if we have an open ball with $r > ...
4
votes
1answer
129 views
Completing $\Bbb R$ when some “divergent” sequences are Cauchy sequences
If I equip $\mathbb{R}$ with the metric
$$
\rho(x,y) := \left|\arctan(x) - \arctan(y)\right|
$$
then sequences like for example $x_n = n$ are Cauchy sequences, so it is clear that $\mathbb{R}$ is ...
0
votes
1answer
181 views
How to prove the strong triangle inequality?
I'm trying to prove that p-adic space is a metric space(also a ultrametric space), but I find it difficult to prove the triangle inequality. So if one can prove the strong triangle inequality, then ...
1
vote
1answer
65 views
Function extending a Lipschitz function
Let $X$ be a metric space with a metric $d$, let $E\subset X$.
We have a function $f:E \rightarrow \mathbb R$ satisfying for some $M>0$:
$$
|f(x)-f(y)|\leq M d(x,y) \quad \text{for } x,y \in E.
$$
...
2
votes
1answer
69 views
What is the relevance of the supremum in this question?
Prove that $d$ is a metric on the set $X$.
$d_u(f,g) = \sup\{|f(x) - g(x)|: x \in I \}, X = C(I)$ the set of all continuous functions from the closed bounded interval $I = [a,b]$ to $\mathbb{R}$
I ...
0
votes
1answer
92 views
Metric space, subsets and true statements
Let $(X, d)$ be a metric space and let $A$ and $B$ be subsets of $X$. Define
$d(A,B) = \inf\{d(a, b) : a \in A, b\in B\}$.
Pick out the true statements.
a. If $A$ and $B$ are disjoint, then $d(A,B) ...
2
votes
2answers
98 views
Metric Of A Graph
The following is question 6 from page 99 of Walter Rudin's Principles Of Mathematical Analysis. I'm having trouble understanding what the metric of the graph might be (which, as far as I can tell, is ...
