Bounded Definition and 514 Threads

Homework Statement Hi everyone. I must show that if f is a continuous function over the complex plane, with limit as z tends to infinity = 0, then f is in fact bounded. The Attempt at a Solution Since f is continuous and lim z --> infinity f(z) = 0, by definition of limit at infinity I know...

Homework Statement If a function f is of bounded variation on [a,b], show it is Riemann integrable Homework Equations Have proven f to be bounded S(P) is the suprenum of the set of Riemann integrals of a partition (Let's say J) s(P) is the infinum of J S(P) - s(P) < e implies f...

a linear operator T: X -> Y is bounded if there exists M>0 such that: ll Tv llY \leq M*ll v llX for all v in X conversely, if i know this inequality is true, is it always true that T: X ->Y and is linear?

Double Integral bounded by Circle? Double integral of (2x-y)dA bounded by circle of radius 2, centered at origin I just need to figure out the limits for my integrals... I am basically lost, can someone show me how to break this up. I tried doing what I did with the previous triangle bound...

I am trying to prove that a nonnegative (so bounded below) harmonic function is constant. I know that Liouville's Theorem states that if a function is bounded both above and below then it is constant. This means it would be sufficient to prove that a harmonic function bounded below is always...

Just working on some practice problems. I missed a couple classes due to sickness and just need some extra help. If you could walk me through how to do these types of problems that would be amazing. Homework Statement Evaluate the volume of the solid bounded by the surfaces (x2 + y2)1/2 =...

Homework Statement Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 9. Homework Equations . . . ? The Attempt at a Solution After drawing out the picture with z=0 I have a line going from 0,9 to 9,0 bounded by the x and y-axis giving me a triangle...

Suppose you're looking at a complex vector space X, and you know that, for some x in X, you have f(x) = 0 for every linear map on X. Can you conclude that x = 0? If so, how? This seems easy, but I can't think of it for some reason. (EDIT: Assume it holds for every CONTINUOUS (i.e...

Homework Statement For integers m and n, let d(m,n)=0 if m=n and d(m,n) = 1/5^k otherwise, where k is the highest power of 5 that divides m-n. Show that d is indeed a metric. Show that, in this metric, the set Z of integers is totally bounded and perfect. The Attempt at a Solution I...

Ok so for a sequence x(n) to be bounded it means |x(n)|<=M but according to by book, if x(n) belongs to some closed interval, say [a,b], x(n) is bounded. That is confusing because say x(n) belonging to [a,b] means a<=x(n)<=b. How can there exist a M such that -M<=x(n)<=M? this means...

Given two functions f_{1}(x) and f_{2}(x) that are differentiable on [a_{1},a_{2}] and \int^{a_2}_{a_1}f_1(x)f'_{2}(x)dx=b_2, how would one calculate \int^{a_2}_{a_1}f_1(x)f'_{2}(x)dx? This is not a homework problem. I saw it on the internet and realized that I did not know where to begin...

Hi everyone, I came across something in my vector calculus textbook (Marsden and Tromba, Edition 5, p. 327) that is confusing me. "A function f(x,y) is said to be bounded if there is a number M>0 such that -M<=f(x,y)<=M for all (x,y) in the domain of f. A continuous function on a closed...

Homework Statement I'm approaching this problem from a different method than conventially shown. Homework Equations if lim=infinity for all M>0, there exists a N such that n>N => {s(n)}>=M The Attempt at a Solution this can be rewritten as: {s(n)} is a sequence. If...

I am asked to prove that any bounded open subset of R is the union of disjoint open intervals. If S = open interval (a,b), I don't really see how this could be the case (there will always be points in S that are not in the union of the disjoint sets).

Homework Statement The region bounded by y=e^-x^2, y=0, x=0 and x=1 is revolved around the y-axis. Find the volume.Homework Equations V = \int_{a}^{b} R^2 dy The Attempt at a Solution We must first express x in terms of y. So we get \ln y = -x^2 x^2 = -\ln y We substitute this to the...

1. Homework Statement [/b] If f has a continuous derivative on [a,b], and if P is any partition of [a,b], show that V(f,P)\leq \intablf'(t)l dt. Hence, Vba\leq\intablf'(t)ldt. Homework Equations Monotone function \subset BV[a,b] \sumf(ti+1)-f(ti) = lf(b) - f(a)l The Attempt at a...

Homework Statement Given a sequence of scalars (cn) and a sequence of distinct points (xn) in (a, b), define f(x) = cn if x = xn for some n, and f(x) = 0 otherwise. Under what condition(s) is f of bounded variation on [a,b]? Homework Equations Vbaf = supp(\Sigmalf(ti) - f(ti-1)l< +inf...

Homework Statement I really got a problem with these products. If Xn is metrizable with dn, and if D(x, y) = suo{di(xi, yi)/i} is the metric which induces the product topology on X = ∏ Xn, show that if Xn is totally bounded for every n (under dn), then X is totally bounded under D...

Homework Statement Show L(X,Y) is a vector space. Then if X,Y are n.l.s. over the same scalar field define B(X,Y) = set of all bounded linear operators for X and Y Show B(X,Y) is a vector space(actually a subspace of L(X,Y) Homework Equations The Attempt at a Solution im not sure if i have...

Hi, If T is a bounded linear operator on a Hilbert space, what can we say about the spectra of T and T* (\sigma(T)=\{\lambda:T-\lambda I is not invertible})?

I am reading about a branch of mathematics which does not allow separable spaces. The author of the text gives the space of functions of bounded variation as an example of a non-separable space, which is fine - except for the fact that he goes on to claim that "this space is relevant to both...

Homework Statement Let X be the integers with metric p(m,n)=1, except that p(n,n)=0. Show X is closed and bounded but not compact. Homework Equations I already check the metric requirement. The Attempt at a Solution I still haven't got any clue yet. Can anyone help me out?

Let (xn) be a seq of real nos and let sn = x1+x2+x3+...+xn / n. prove that if if xn is bounded and monotone, then sn is also bdd and monotone. How can i got about this one.. ? I got it in the test today and i couldn't figure it out. only hint i could think of is how do i prove if xn...

Homework Statement try ∫∫G x^2 dA ;value is the region bounded by the ellipse 9x^2+4y^2=36 Homework Equations The Attempt at a Solution i think i have to change the variables to polar coordinate or U,V function but i have no idea.

The area of the region bounded by the curve y = 3x^2-3 , the y-axis, x-axis, and the line x = 2 is equal to so far i ve managed to draw the graph i m getting a value of 9 is that correct

Hi, Suppose f is an entire function such that f(z) = f(z+2pi) = f(z+2(pi)i) for all z E C. Use Liouville's theorem to show that f is constant. Obviously I need to show that the function is bounded but I'm unsure of how to approach it. The hint is: Consider the restriction of f to the...

Homework Statement Evaluate the triple integral of the function f(x,y,z) = x over the volume bounded by the surfaces 2x + 3y + z =6,x=0,y=0,z=0. Homework Equations The Attempt at a Solution See figure attached for my attempt. I sketched the volume bounded by the surfaces...

Homework Statement Let f be a function and p\in . Assume that a\leqf(x)\leqb near p. Prove that if L= lim f(x) as x-->p Then L\in [a,b] The Attempt at a Solution I want to say that because f(x) is bounded by [a,b] that automatically implies that the Limit L is also bounded by...

I don't fully understand this theorem: Let Q and Q' be two rectangles in R^n. If F: R^n -> R is a bounded function that vanishes outside Q intersect Q', then integral of f over q is equal to the integral of f over Q'. When it says that the function vanishes outside of Q intersect Q', does...

Homework Statement Forgive my lack of LaTeX, not learned how to use it yet. Anyway, the problem is: Use the inequalities 1/(n+1) < ln(n+1) - ln(n) < 1/n to show that the sequence {xn} from n=1 to infinity defined by xn = 1 + 1/2 + ... + 1/n - ln(n) is strictly decreasing and bounded below...

Homework Statement Give an example of a function f that is differentiable on [0,1] but its derivative is not bounded on [0,1] Homework Equations The Attempt at a SolutionOk, I know that the derivative f' cannot be continuous, because then it would be bounded on [0,1]. I also know that it...

1. Let R be the region bounded by the curves y = x2 and y = x + 2. (a) Sketch the region R and label the points of intersection between the two curves. (b) Suppose we rotate R about the x-axis. Compute the volume of the resulting solid. (c) What is the volume of the solid obtained by rotating...

Homework Statement Ok so I'm given that we have some function from R to R, that is continuous on all of R. I am asked if it is possible to find some BOUNDED subset of R such that the image of the set is not bounded. The professor gave the hint: look at closures. Homework Equations...

Homework Statement My mind is blown. You'd think there would be some number which 1/1 + 1/3 + 1/4 + ... stays below, but I guess there isn't. However, before I believe this, I need one part of my book's proof clarified. Homework Equations Theorem I. Suppose that un ≥ 0 for every n...

Homework Statement Suppose M is a metric space and A \subseteq M. Then A is totally bounded if and only if, for every \epsilon >0, there is a finite \epsilon-dense subset of A. Homework Equations The Attempt at a Solution I have already done the \Rightarrow but need to verify...

Homework Statement Consider a 2d potential problem for a region bound by 4 planes x=-1/2 x=1/2 y=0 y=1 There are no charges inside the bounded region. The boundaries at y=0 and y=1 are held at zero potential. The potential at the boundaries x=-1/2 and x=1/2 is given by...

Homework Statement Let (X,d), (Y,p) be metric spaces, and let f,fn: X -> Y with fn->f uniformly on X. Show that D(f)c the union of D(fn) from n=1 to n=infinity, where D(f) is the set of discontinuities of f. Homework Equations The Attempt at a Solution Ok, so this looks pretty...

Homework Statement Prove that the following function is continuous and bounded in R+ Homework Equations f(x) = 1 + \int_0^x e^{-t^2} f(xt) dt \qquad \forall x \geq 0 The Attempt at a Solution I thought of using Taylor Formula, but the integral is t instead of x, so now i have...

Hey all, We were discussing bounded and unbounded sets in class, and looking over my notes, I see that I have some trouble understanding the concept. Here are three examples that our professor gave us: Set A = {x\inR | |x| <10} Set A = {x\inR | x<10} A\subseteqZ s.t. x~y iff x|y Set A =...

Homework Statement Prove that a bounded subset of R is totally bounded. Homework Equations The Attempt at a Solution Fix E > 0. Let A be subset of R, x be contained in A, and B(E/2, a) where E/2 is the radius of the ball and a is the center. Assume that B(E/2, a) is closed...

"if a vector field has only nondegenerate zeros then the number of zeros is bounded" With no idea how to show that without using Poincare-Hopf Theorem. Any proof possible without using any concept from algebraic topology?

Homework Statement Prove that SU(n) is closed and bounded Homework Equations The Attempt at a Solution So in order to prove this, I first mapped SU(n) to be a subset of R^{{2n}^2}. To prove the closed portion, I tried mapping a sequence in SU(n) to a sequence in R^{{2n}^2}. However, I...

Here's the problem: Let f:D-R and c in R be and a accumulation point of D, which is a subset of R. Suppose that a<=f(x)<=b for all x in D, x not equal to c, and suppose that limx\rightarrowc f(x) = L. Prove that a<=L<=b. I'm having trouble here. I've tried to prove by contradiction, by...

Homework Statement Let (sn) be a sequence in R that is bounded but diverges. Show that (sn) has (at least) two convergent subsequences, the limits of which are different. Homework Equations The Attempt at a Solution I know that a convergent subsequence exists by...

Homework Statement Asssume (an) is a bounded sequence with the property that every convergent subsequence of (an) converges to the same limit a. Show that (an) must converge to a. Homework Equations The Attempt at a Solution If the subsequence converges to a we have , we have...

Homework Statement Let A = {x: x^2 + 3x + 2 <0}. Prove that this set is non-empty and bounded above. What is the least upper bound? Is it bounded below? The Attempt at a Solution Well, solving for the zeros and understanding that between the zeros, we satisfy our values of x's, I have...

Homework Statement let |e-x-e-y| be a metric, x,y over R. let X=[0,infinity) be a metric space. prove that X is closed, bounded but not compact. Homework Equations The Attempt at a Solution there is no problem for me to show that X is closed and bounded. but how do I prove...

Homework Statement Show that a sequence {a_n} is bounded if and only if there is an interval [c, d] such that {a_n} is a sequence in [c,d]. Homework Equations A sequence {a_n} is bounded provided that there is a number M such that |a_n| <= M. The Attempt at a Solution...

Question about the "bounded" metric Homework Statement I'm trying to show that the topology induced on \mathbb R by the bounded metric \sigma(x,y) = \left| \frac{x}{1+|x|} - \frac{y}{1+|y|} \right| is equivalent to that induced by the standard Euclidean metric d(x,y) = |x-y|. Homework...

Take the discreet metric on an infinite set A. I understand that its closed (because it contains all of its limit points), but I don't understand why its bounded and why its not compact. Also, when they say "an infinite set A" do they mean a set that extends to infinite (say, [1,n] for...