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facenian
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Is there an easy example of a closed and bounded set in a metric space which is not compact. Accoding to the Heine-Borel theorem such an example cannot be found in ##R^n(n\geq 1)## with the usual topology.
From Wikipedia:facenian said:Is there an easy example of a closed and bounded set in a metric space which is not compact. Accoding to the Heine-Borel theorem such an example cannot be found in ##R^n(n\geq 1)## with the usual topology.
facenian said:Is there an easy example of a closed and bounded set in a metric space which is not compact. Accoding to the Heine-Borel theorem such an example cannot be found in ##R^n(n\geq 1)## with the usual topology.
I think it will be hard to beat this example on simplicity basis.fresh_42 said:From Wikipedia:
A simple counterexample yields the discrete metric on an infinite set of ##X##. The discrete metric is defined by
##d (x, x) = 0##,
##d (x, y) = 1## for ## x \neq y\; , \; x, y \in X ##
In this metric, each subset of ##X## is closed and bounded, but only the finite subsets are compact.
Further counterexamples are all infinite-dimensional normalized vector spaces.
No. It is the closed unit disk. If it were open then a Cauchy sequence of points that converges to a point of distance 1 to the origin will not converge.facenian said:I think it will be hard to beat this example on simplicity basis.
In the case of @lavinia 's example I believe the disk should be an open unit disk.
Thank you guys, it's been very helpful
Of course!, I'm sorry I was thinking what must happen for it not to be compact and responded on the contrary. Thanks again.lavinia said:No. It is the closed unit disk. If it were open then a Cauchy sequence of points that converges to a point of distance 1 to the origin with not converge.
facenian said:Of course!, I'm sorry I was thinking what must happen for it not to be compact and responded on the contrary. Thanks again.
Can you describe the topology a little more?mathman said:Another simple example Hilbert space. The unit ball is closed and bounded but not compact. A particular Hilbert space would be all infinite series, where the sum of the squares is finite.
lavinia said:Can you describe the topology a little more?
Yes, there are infinite nets without finite subnets, which is an equivalent to compactness: every e-net has a finite subnet. I think also the sequence ##e_i:= \delta^i_j## has no convergent subsequence, since ##||e_i-e_j||= \sqrt 2 ##Math_QED said:Probably @mathman refers to the space ##\mathcal{l}^2 = \{(a_n)_n \in \mathbb{K}^\mathbb{N}\mid \sum |a_n|^2 < \infty\}##, with the norm induced by the innerproduct ##\langle (a_n)_n, (b_n)_n\rangle:= \sum a_n b_n##.
I.e., ##\Vert (a_n)_n \Vert = \sum |a_n|^2 < \infty##
You take the topology that is derived from the norm.
WWGD said:Yes
Yes, there are infinite nets without finite subnets, which is an equivalent to compactness: every e-net has a finite subnet. I think also the sequence ##e_i:= \delta^i_j## has no convergent subsequence, since ##||e_i-e_j||= \sqrt 2 ##
When I first was exposed to Hilbert space, it was only infinite dimensional, i.e. the infinite dimensional extension of Euclidean space.mathwonk said:@mathman, I know you meant this, but technically you need to say infinite dimensional Hilbert space. Indeed a normed space is locally compact iff it is finite dimensional, Riesz's theorem. Dieudonne', Foundations of modern analysis, page 109.
Does that mean in your world there is no zero operator or at least no restriction of the codomain to the image? I think from a categorial point of view to include finite dimensional Hilbert spaces make absolute sense.mathman said:When I first was exposed to Hilbert space, it was only infinite dimensional, i.e. the infinite dimensional extension of Euclidean space.
A compact subspace in a metric space is a subset of a metric space that is both closed and bounded. This means that the subspace contains all of its limit points and can be contained within a finite distance.
Compactness and connectedness are two different properties of a subset in a metric space. A compact subset is closed and bounded, while a connected subset is not separated into two or more disjoint sets. In other words, a compact subspace can be contained within a finite distance, while a connected subspace cannot be split into two or more separate parts.
No, a compact subspace must be both closed and bounded. If a subspace is unbounded, it cannot be contained within a finite distance and therefore cannot be considered compact.
The Heine-Borel theorem states that a subset in a metric space is compact if and only if it is closed and bounded. In other words, a subset is compact if and only if it satisfies the definition of a compact subspace.
Some examples of compact subspaces in a metric space include a closed interval [a,b] in the real numbers with the standard metric, a closed disk in the plane with the Euclidean metric, and a closed and bounded set of points in a Euclidean space with the standard Euclidean metric.