Tagged Questions
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
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 $\epsilon$
...
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 ...
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 > ...
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 ...
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 ...
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 ...
1
vote
0answers
123 views
A metric space is path connected and countable then it is complete
I have to show that if a metric space is path connected and countable then it is complete. I'm pretty lost where to start this at all. I have the basic definitions of complete, path-connected, compact ...
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
...
0
votes
0answers
134 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] ...
0
votes
0answers
69 views
Giving $\mathbb{R}^n$ finite diameter
Endow $\mathbb{R}^n$ for ($n\in\mathbb{Z}_{\ge1}$) with the (standard) Euclidean metric $d$. Let $f\colon[0,\infty)\to[0,\infty)$ be a subadditive mapping and satisifies $f(x)=0\iff x=0$ for all ...
