All Questions
6
questions
2
votes
1
answer
73
views
Does the supremum norm $\|p\|_{A}$ depend continuously on subsets $A\subset\mathbb{C}$ with respect to the Hausdorff distance?
Consider the space $\mathcal{K}$ of all non-empty compact subsets of $\mathbb{C}$. One can show that the Hausdorff distance defined by
$$h(X,Y)=\max\bigg\{\sup_{x\in X}\inf_{y\in Y}|x-y|,\sup_{y\in Y}\...
- 3,376
-1
votes
2
answers
101
views
Proving the principal argument not continuous using standard metrics [closed]
Let $\operatorname{Arg}: \Bbb{C} \setminus \{0\} \to\Bbb{R}$ be the principal value of the argument, taking values in $(−\pi, \pi]$. Using the standard metrics on $\Bbb{C} \setminus \{0\}$ and $\Bbb{R}...
- 19
1
vote
1
answer
171
views
Dense subsets of $C^0(B_1,\mathbb C)$.
Let $B_1 := \{z \in \mathbb C : |z| \le 1\},$ and let $C^0(B_1,\mathbb C)$ be the space of continuous complex-valued functions on $B_1$ equipped with the uniform convergence topology.
Listed below ...
- 133
1
vote
1
answer
25
views
Denseness of the image of a certain continuous function from real line to the product of two copies of unit circles
Let $a,b \in \mathbb R$ such that $a/b$ is irrational . Consider the continuous function $f : \mathbb R \to S^1 \times S^1$ as $f(t)=(e^{2\pi iat } , e^{2\pi ibt}) , \forall t \in \mathbb R$ ; then is ...
user456828
1
vote
1
answer
313
views
How to prove this metric function is continuous?
Suppose that X and Y are metric spaces with distance functions dX and dY and define a function
d : (X × Y ) × (X × Y ) → [0, ∞) by
d((x, y),(x',y')) = dX(x, x') + dY (y, y').
Check that d defines ...
- 23
1
vote
3
answers
78
views
Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?
Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
user228169