Prove that if a_{n} \to Aa_{n} \to A for some real AA then a_{n}^{2} \to A^{2}a_{n}^{2} \to A^{2}

Let A \in \mathbb{R} and let (a_{n}) be a sequence of reals convergent to A. We have |a_{n}^{2} – A^{2}| = |a_{n} – A||a_{n} + A|. I am stuck at majorizing |a_{n} + A|. I know that convergent sequences are bounded, but, that kind of bounds, which takes the form \max \{ |a_{1}|, \dots, |a_{N-1}|, … Read more

Prove that TT is bounded if ⟨Tx,y⟩=⟨x,T∗y⟩\langle Tx, y \rangle = \langle x, T^*y \rangle

Suppose T:L2(Rd)→L2(Rd) is a linear operator, and there exists T∗:L2(Rd)→L2(Rd) such that ⟨Tx,y⟩=⟨x,T∗y⟩ for all x,y∈Rd. Prove that T is bounded. I have no idea how to approach this problem. I think it says that T is “invertible” if it is bounded, which makes sense. I have no idea how to prove it though. Right … Read more

Polygamma reflection formula

How does one prove the polygamma reflection formula: ψ(n)(1−z)+(−1)n+1ψ(n)(z)=(−1)nπdndzncotπz Do we have to invoke the power of contour integration and kernels? I searched the web but the proof was to be found nowhere. Answer You just need to prove the reflection formula: ψ(1−z)−ψ(z)=πcot(πz) then differentiate it multiple times. In order to prove (1), let’s start … Read more

Angle-doubling map is mixing

Let T:S1→S1x↦2x be the angle-doubling map on the circle. We know that this transformation is ergodic. We want to prove that is mixing. I have to show that lim for every f,g\in L^2. The idea is to consider e^{imx} which spans L^2. Then I can’t go on. Answer Two standard arguments: (1) In order to … Read more

Understanding L’hospital

I have some trouble understanding L’Hospital’s rule. Let’s say I am given the function f(x)=2x5x+6×2 I am interested in lim. (1) \lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} \frac{2}{5+12x} = \frac{2}{5} (2) \lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} \frac{2}{5+12x} = \lim\limits_{x \to 0} \frac{0}{12} = 0 Where’s the mistake? Answer The second … Read more

Existence of a metric space where each open ball is closed and has a limit point

Show that there exists a metric space in which every open ball is closed and contains a limit point. I think that the space {1n∣n∈N,n>0}∪{0} with the standard Euclidean metric is an answer, but it is not true open ball with center 1 is closed. Answer Z with the p-adic metric d(m,n)=p−νp(m−n) has the property … Read more

How do I find the exact solution to the boundary value problem y″y” = 4y’ + y + 2 − 8x − x^ {2} , y(0) = 0y(0) = 0 and y(4) = 16 y(4) = 16?

I am approaching this question by trying to guess the general solution to the boundary value problem. However I haven’t come up with one. Can someone explain how to solve this question please? Answer I will go through the details if you like, but the general solution is achieved by finding the complementary and particular … Read more

Do we need to have a subsequence such that limk→∞‖\lim_{k\to\infty}\left\|x_{n_k}\right\|=\liminf_{n\to\infty}\left\|x_n\right\|?

Let (X,\left\|\;\cdot\;\right\|) be a normed space and (x_n)_{n\in\mathbb{N}}\subseteq X. Can we prove that there is a subsequence \left(x_{n_k}\right)_{k\in\mathbb{N}}\subseteq(x_n)_{n\in\mathbb{N}} such that \lim_{k\to\infty}\left\|x_{n_k}\right\|=\liminf_{n\to\infty}\left\|x_n\right\|\;? Answer Yes, and this holds more generally for real sequences. In fact, the limit inferior of a sequence can be defined as the least limit point of any subsequence of the sequence. To prove … Read more