Compact Definition and 325 Threads

Homework Statement The Dirichlet Prime Number Theorem indicates that if a and b are relatively prime, then the arithmetic progression A_{a,b} = \{ ...,a−2b,a−b,a,a+b,a+2b,...\} contains infinitely many prime numbers. Use this result to prove that Z in the arithmetic progression topology is not...

Theorem: Let S be a compact subset of ℝ^n. Then S is closed. Before looking at the book I wanted to come up with my own solution so here is what I've thought so far: Fix a point x in S. Let Un V_n (union of V_n's...) be an open covering of S, where V_n=B(x;n). We know that there is a...

Hi, I'm struggling to understand this concept. I think the term probably comes from functional analysis and I don't know any of the terms in that field so I'm having trouble understanding the meaning of what a compact linear operator is. I posted this in linear algebra because I'm reading...

Homework Statement So I am trying to understand this proof and at one point they state that an arbitrary compact subset of a Banach space, or a completely metrizable space is the subset of a finite set and an arbitrary convex neighborhood of 0. I've been looking around and can't find anything...

Homework Statement Hi all, I am struggling with getting an intuitive understanding of linear normed spaces, particularly of the infinite variety. In turn, I then am having trouble with compactness. To try and get specific I have two questions. Question 1 In a linear normed vector space, is...

Homework Statement So I had a homework problem which was to show that a certain function space was compact via Ascoli-Arzela Theorem. I was okay with doing this, accept, it appears the corresponding uniformly convergent sequence I found in any infinite set need not converge to something in the...

Homework Statement Let w_1,...,w_n be a set of n-linearly independent vectors in \mathbb{R}^n. Define an equivalence relation \sim by p\sim q \iff p-q=m_1w_1+...+m_nw_n for some m_i \in \mathbb{Z} Show that \mathbb{R}^n / \sim is Hausdorff and compact and actually homeomorphic to (S^1)^n...

I just want to know the difference between fluorescent strip and compact fluorescent lamp... Thanks.

So, I'm working a bit through munkres and I came across this problem Show that every locally compact Hausdorff space is regular. So, I think I've solved it, but there is something confusing me. I initially said that if X is locally compact Hausdorff, it has a 1-point compactification, Y...

Consider the sequence $\{f_n\}$ of complex valued functions, where $f_n=tan(nz)$, $n=1,2,3\ldots$ and $z$ is in the upper half plane $Im(z)>0$. I want to show two facts about this sequence: 1) it's uniformly locally bounded: for every $z_0=x_0+iy_0$ in the upper half plane, ther exist...

eigenvalues of a compact positive definite operator! Let A be a compact positive definite operator on Hilbert space H. Let ψ1,...ψn be an orthonormal set in H. How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where λ1≥λ2≥λ3≥... be the eigenvalues of A in decreasing order. Can...

Hi, In Baby Rudin, Thm 3.6 states that If p(n) is a sequence in a compact metric space X, then some subsequence of p(n) converges to a point in X. Why is it not the case that every subsequence of p(n) converges to a point in X? I would think a compact set would contain every sequence...

So this is more so a general question and not a specific problem. What exactly is the diefference between closed and boundedness? So the definition of closed is a set that contains its interior and boundary points, and the definition of bounded is if all the numbers say in a sequence are...

Homework Statement Let A,B be two disjoint, non-empty, compact subsets of a metric space (X,d). Show that there exists some r>0 such that d(a,b) > r for all a in A, b in B. Hint provided was: Assume the opposite, consider a sequence argument. Homework Equations N/A The Attempt...

Q1. if A is a subset of X, choose the topology on X as {∅,U|for every U in X that A is a subset of U}. Then is this topology a locally compact space? Q2. X=[-1,1], the topology on X is {∅, X, [-1, b), (a,b), (a,1] | for all a<0<b}. How to prove every open set (a,b) in X is NOT locally compact...

I am currently reading a paper discussing the convexity of the image of moment maps for loop groups. In particular, if G is a compact Lie group and S^1 is the circle, the paper defines the loops group to be the set of function f: S^1 \to G of "Sobolev class H^1 ." Now in the traditional...

Homework Statement Basically, prove the Extreme Value Theorem. "If f is a continuous function over the interval [a,b] then f reaches a max and a min on that interval." Homework Equations In this case they're more like definitions and things I have proved so far. Intervals are...

what is the the carrier plasma dispersion effect?

The first commercial CD release was 30 years ago yesterday, in Japan: Billy Joel's "52nd Street." http://www.cnn.com/2012/09/28/tech/innovation/compact-disc-turns-30/index.html?hpt=hp_bn5 I waited two and a half years before buying my first CD player, for prices to come down a bit and a decent...

How small can a maglev propulsion system be? What's the relationship between the size/strength of the magnets and the load capacity of a maglev vehicle?

Homework Statement I need to prove that a closed ball(radius r about x0) is closed and bounded. The same goes for a sphere(radius r about x0). Homework Equations The Attempt at a Solution How does one go about proving something is closed and bounded? My book is not very helpful...

Spivak's proof of "A closed bounded subset of R^n is compact" Hi guys, I'm currently taking a differential geometry course and decided I would read Spivak's Calculus on Manifolds, and then move on to his Differential Geometry series. There's a proof in here that feels unjustified to me, so...

I was just googling around and I came across this problem. Let (X,d) be a metric space. Let (An)n \in N be a sequence of closed subsets of X with the property An \supseteq An+1 for all n \in N. Suppose it exists an m \in N such that Am is compact. Prove that \bigcapn\in NAn is not empty...

Hi, In the middle of the article about Franciscus Vieta, here: http://en.wikipedia.org/wiki/Franciscus_Vieta I see an infinite product as an expression for Pi: 2 * 2/2^(1/2) * 2/(2+(2^(1/2))^(1/2) * ... I was wondering, how this could be written in compact form using math notation please?

Homework Statement Let H be an \infty-dimensional Hilbert space and T:H\to{H} be an operator. Show that if T is compact, bounded and has closed range, then T has finite rank. Do not use the open-mapping theorem. Let B(H) denote the space of all bounded operators mapping H\to{H}, K(H) denote...

This device was recently purchased at a hardware store, and it has a wattage of 1.1kW but can fit inside a pocket with dimensions of approximately 12cm x 6cm x 3cm. Does it use a neodymium-iron-boron alloy with 44 times the magnetic energy density of iron as the core of a transformer, an...

Homework Statement A compact disc speeds from rest to 5200 rpms in 620 rad. Diameter is 5.0cm -how many revolutions did it make in this time -what is the angular acceleration in rad/s^2 -how long does ti take to reach this speed Homework Equations The Attempt at a Solution i was able to find...

Compact --> bounded In lecture 8 of Francis Su's Real Analysis online lecture series, he has a proof that a compact subset of a metric space is bounded: Given a metric space (X,d), if A is a compact subset of X, then every open cover of A has a finite subcover. Let B be a set of open balls of...

Homework Statement Let [e_{j}:j\in N] be an orthonormal set in a Hilbert space H and \lambda_{k} \in R with \lambda_{k} \rightarrow 0 Then show that Ax=\sum_{j=1}^{\infty} \lambda_{j}(x,e_{j})e_{j} Defines a compact self adjoint operator H \rightarrow H The Attempt at a Solution...

Homework Statement {1,1/2,2/3,3/4,4/5...} is this set compact. The Attempt at a Solution I think this set is compact because it contains its cluster point which is 1. is this correct?

Homework Statement Decide whether the following propositions are true or false. If the claim is valid supply a short proof, and if the claim is false provide a counterexample. a) An arbitrary intersection of compact sets is compact. b)A countable set is always compact. The Attempt at a...

Homework Statement X is a Banach space S\in B(X) (Bounded linear transformation from X to X) T\in K(X) (Compact bounded linear transformation from X to X) S(I-T)=I if and only if (I-T)S=I The question also asks to show that either of these equalities implies that I-(I-T)^{-1} is compact...

Homework Statement This problem is in Analysis on Manifolds by Munkres in section 25. R means the reals Suppose M \subset R^m and N \subset R^n be compact manifolds and let f: M \rightarrow R and g: N \rightarrow R be continuous functions. Show that \int_{M \times N} fg = [\int_M f] [...

Homework Statement Let S be the set of all cumulative distribution functions (cdf), as defined in basic statistics (defined below). Along with a metric d, (S,d) will defined a metric space. My goal is (with any metric) to make a compact subset from this set S. Does anyone have any...

A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. Does not this imply that every open set is compact. Because let F is open, then F= F \bigcup ∅. Since F and ∅ are open , we obtained a finite subcover of F. Am I missing something here?

Let $X$ be a compact and convex subset of $\mathbb{R^2}$ Let $a^1, a^2 \in X$ such that $a^j = (a^j_1, a^j_2)$, $j=1,2$ Is $c= \sum_{i=1}^2 \mathbb{I}_{ i=j} a^j_i \in X \quad ?$

I need some help in understanding exactly the following definition (any links to sources will be great). What is boundedly weakly sequentially compact subset in the compact Hausdorf space with a regular Borel measure? What in this case will be the weak topology? Thanks!

Hi, Why R mod 2pi is a Compact Manifold? Isn't this like a real line which is not compact? How should we prove it using a finite sub-cover for this manifold? bah

Hi there. I'm looking at Poincare duality, and there's something extremely wrong with the way I'm looking at one or more of the concepts and I need to figure out which. When dealing with non-compact manifolds, you can fix Poincare duality by looking at something called "cohomology with...

In Theorem 26.7 of Munkres' Topology, it is proved that a product of two compact spaces is compact, and I think the author seems to (rather sneakily) use the choice axiom without mentioning it... Could anyone tell me if this is indeed the case? I don't have a problem with the choice axiom, but...

Hi Suppose (X,d) is a bounded metric space. Can we extend (X,d) into (X',d') such that (X',d') is compact and d and d' agree on X? ( The reason for asking the question: To prove a theorem in Euclidean space, I found it convenient to first extend the bounded set in question to a compact one (...

I'm looking at prop 19.5 of Taylor's PDE book. The theorem is: If M is a compact, connected, oriented manifold of dimension n, and a is an n-form, then a=dB where B is an n-1 form iff the ∫a over M is 0. I'm trying to understand why a=dB implies ∫a = 0. If M has no boundary, than this...

Hello physicsforum - I recently began self-studying real analysis on a whim and I have run into some trouble understanding the idea of compactness. I have no accessible living sources near me and I have yet to find a source that has explained the matter clearly, so I shall turn here for...

Homework Statement Let X be a compact and locally connected topological space. Prove that by identifying a finite number of points of X, one gets a topological space Y that is connected for the quotient topology. Homework Equations The components of a locally connected space are open...

Counterexample "intersections of 2 compacts is compact"? Hello, I'm looking for a counterexample to "If A and B are compact subsets of a topological space X, then A \cap B is compact." It's not for homework. I found one online, but it talked about "double-pointed" things which I didn't...

I came across this proof and have a question about the bolded portion: Consider the following objection to the bolded: In order for \mathcal{G} to be an open cover of K its sets must contain all of the points of K. The sets of \mathcal{G} are B_r(p) for some fixed p, and so as r gets...

I'm trying to prove that if a function is continuous on [a,b] and smooth on (a,b) then there's a point x in (a,b) where f'(x) exists. The definition of smoothness is: f is smooth at x iff \lim_{h \rightarrow 0} \frac{ f(x+h) + f(x-h) - 2f(x) }{ h } = 0 . I'm starting with the simpler case...

Are compact florescent light bulbs know to be strong sources of ultrasounic noise? I've been playing with a "Marksman Ultrasonic Diagnostic Tool", which presents ultrasounds as audible sound. It isn't a scientific instrument, so it doesn't give a decibel reading. It is used to detect leaks...

Suppose X is compact and that f:X→ℝ, with the usual metric. In addition suppose that {x: f(x)>a} is open for every a ∈ ℝ. I need to show that there is always an x ∈ X such that f(x) equals the supremum of the range of f, or I need to provide a counter example. For what it's worth we also...

Are there any down to Earth examples besides the empty set? Edit: No discrete metric shenanigans either.