New answers tagged metric-spaces
1
vote
What's the most logical decay profile *in a round space* of a force whose decay is approximated locally by the square of distance?
The mathematical formulation: A Riemannian manifold has a divergence operator acting on vector fields of flows. The divergence at a point is the limit of the outward flow over n-1-sphere containing ...
- 6,020
1
vote
Accepted
Bracketing number vs covering number
If two reals $x$ and $y$ are of different signs, then $|x +y| \le \max(|x|, |y|)$. Now, since $g - l_i \ge 0$ and $g - u_i \le 0,$ then, pointwise, $$ \frac{|g- l_i + g - u_i|}{2} \le \frac{\max( | g- ...
- 11.2k
14
votes
Accepted
Difficulties with Set-Notation in Taxicab Geometry
…the set of points $P$ that is the answer is dependent on the distance between $A$ and $B$, thus I am to observe some property of the relationship between the sum of circles, which hardly makes sense ...
- 1,321
12
votes
Difficulties with Set-Notation in Taxicab Geometry
As @XanderHenderson notes in his excellent answer, this is not about adding sets, it's about adding distances - in this case taxicab distances. It's not about circles either.
One way to think of the ...
- 109k
12
votes
Difficulties with Set-Notation in Taxicab Geometry
This is not about "adding sets" (though such a notion does exist—you might search the term "Minkowski sum"). Rather, the set is defined by an equation, and you are being asked to ...
- 34k
0
votes
Weak topology on an infinite-dimensional normed vector space is not metrizable
We can prove it by contradiction. Suppose that the weak topology is induced by a metric $d$. Let $U_n = \{d(0, x) < 1/n\}$. Then $U_n$ is unbounded in the norm topology. Take $x_n \in U_n$ such ...
- 430
0
votes
Is the topology that has the same sequential convergence with a metrizable topology equivalent as that topology?
No. This is not true. Schur's lemma says,
Lemma: In $l^1$, a sequence converges in the weak topology if and only
if it converges in the norm topology.
For example, see Proposition 5.2 in Conway's ...
- 430
2
votes
The isometry classes of compact metric spaces form a set?
Any compact metric space can be embedded in $[0,1]^{\mathbb{N}}$, so indeed the isometry classes form a set (precisely: the set of isometry classes of subspaces of $[0,1]^{\mathbb{N}}$).
- 27.9k
1
vote
Properties of quotient pseudometric and image geometry for a continuous surjection between parameter spaces
I just wanted to explore the conept of pushforward of a pseudometric, and this is the only question I found about it.
If $f:X\to Y$ is a surjection and $d_X$ is a pseudometric on $X$, then the ...
- 17.3k
1
vote
Does every smooth manifold admit a generic Riemannian metric?
In the MathOverflow post you link to, equations (2.15), (2.16), (2.17) are at each point a linear system of equations for the tensors $H$ and $F$. The vector space of all such tensors $(H,F)$ has ...
- 12.2k
-3
votes
Does every smooth manifold admit a generic Riemannian metric?
A covariant derivative $\mathbf{ \mathbb \nabla}$ defined freely by a connection field $x\to \Omega(x)$, acting on vector fields $x\to \mathbf X(x)$, locally spanned by partial derivatives $(x,i) \to (...
- 6,020
3
votes
Accepted
$f+g,f-g,f\cdot g, \frac{f}{g}$ are continuous. ("Topology Second Edition" by James R. Munkres.)
Why is the proof that sums, differences, products, and quotients of continuous real-valued functions are continuous placed in the section on metric spaces, even though it seems that it could just as ...
- 96.7k
2
votes
Accepted
What is the optimal strategy to locate a hidden point within 1ft across a large area using only distance-change feedback rounded to the nearest 5 ft?
Let the first two moves travel along an equilateral triangle of maximal size (from the origin, along $15^\circ$ North of East and along $15^\circ$ East of North, until these reach the boundary of the ...
- 73.3k
3
votes
What is the optimal strategy to locate a hidden point within 1ft across a large area using only distance-change feedback rounded to the nearest 5 ft?
Given an unknown but fixed $t = (t_x, t_y)$, let $p_i = (x_i, y_i)$ be your current position at step $i$, so $p_0 = (0,0)$. Let $$\delta_i = ||p_i - t|| - ||p_{i-1} - t||, \quad i \ge 1 \tag{1}$$ ...
- 151k
3
votes
Accepted
Find the distance between these two sets.
The argument is fine. I think that it could be presented in a way that is a bit more clear—you are, apparently, claiming that the distance between the sets is zero, but you don't actually say that ...
- 34k
7
votes
Proof that categories of Hausdorff spaces and metric spaces are not cartesian closed
There are existing results in the literature which answer some of your questions. For instance, I believe that it was known quite early that $\mathrm{Haus}$ is not Cartesian closed. Right now I'd like ...
- 18.4k
1
vote
Rudin $7.14$ linked to $7.9$ why does Rudin make boundedness compulsory in uniform convergence?
Rudin does not make boundedness compulsory for uniform convergence.
What he writes is that "a sequence $\{f_n\}$ converges uniformly to $f$ with respect to the metric of $\mathscr C(X)$ if and ...
- 8,846
1
vote
Accepted
Prob. 15, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: The subspace $\mathscr{C}(I,\mathbb{R})$ of $\mathbb{R}^I$ with uniform metric is separable
Is $\mathscr{F}$ countable?
Yes. The argument in your postscript is on the right track. I cannot see an obvious bijection between $S = I \cap \mathbb{Q}$ and $\{0,1\} \cup S \cup S^2 \cup S^3 \cup ......
- 3,942
4
votes
Accepted
Proof that categories of Hausdorff spaces and metric spaces are not cartesian closed
This answer shows that all mentioned categories are not cartesian closed.
$\mathbf{Met}$ is not cartesian closed
For $r \geq 0$ consider the metric space $T_r$ which has two points $0,1$ with distance ...
- 186k
Top 50 recent answers are included