Recent content by ELESSAR TELKONT

I tried with contradiction argument on this: Let B_{x,\epsilon_{1}}^{\rho_{1}}\times B_{y,\epsilon_{2}}^{\rho_{2}} no open in (X\times Y,\rho_{3}). Then there is some z_{0}\in B_{x,\epsilon_{1}}^{\rho_{1}}\times B_{y,\epsilon_{2}}^{\rho_{2}}=C such that for every \delta>0...

Well, as I have said in my first post this can be reduced to think about balls. I'll go more explicit about this. As Halls of Ivy said if z\in A\times B, by definition of cartesian product there are some x\in A,\, y\in B such that z=(x,y). Since the factor sets are open in their metric spaces...

Homework Statement This is not really coursework. Instead, this is some sort of curiosity and proposition formulation rush. Then the initial questions are that if this is a valid result that is worth to be proven. Let X,Y be metric spaces and X\times Y with another metric the product metric...

Why I can't see that! that's another version of the Young's inequality. thanks for that illuminating idea.

The result for \mathbb{R}^{n} is Let 1\leq p,q that satisfy p+q=pq and x,y\in\mathbb{R}^{n}. Then \begin{align} \sum_{k=1}^{n}\left\vert x_{k}y_{k}\right\vert\leq\left(\sum_{k=1}^{n}\left\vert x_{k}\right\vert^{p}\right)^{\frac{1}{p}}\left( \sum_{k=1}^{n}\left\vert...

The problem I have is not the proof itself, but the convergence of the LHS of the inequality. How can I prove it is the question. Obviously in \mathbb{R}^{n} you don't need to check any convergence, then you have no manner to parallel that part of the proof. In other words: I have followed the...

Homework Statement Let 1\leq p,q that satisfy p+q=pq and x\in\ell_{p},\, y\in\ell_{q}. Then \begin{align} \sum_{k=1}^{\infty}\left\vert x_{k}y_{k}\right\vert\leq\left(\sum_{k=1}^{\infty}\left\vert x_{k}\right\vert^{p}\right)^{\frac{1}{p}}\left( \sum_{k=1}^{\infty}\left\vert...

Thanks. I have not thought in that manner previously. Here is my proof of the implication I had problems with: Let the equality is true. Suppose that f is not inyective. Then there exists b\in \Im f such that there are at least two x_{1},x_{2}\in A such that are preimages of b vía f. Let C...

Homework Statement Let A, B be sets, C,D\subset A and f:A\longrightarrow B be a function between them. Then f(C\cap D)=f(C)\cap f(D) if and only if f is injective. Homework Equations Another proposition, that I have proven that for any function f(C\cap D)\subset f(C)\cap f(D), and the...

How, give me a hint

Homework Statement A charge distribution has the shape of a very large disk of thickness 3d. This disk has three layers of uniform densities \rho_{1},\rho_{2},\rho_{3} and thickness d each one. Within the layer with density \rho_{2} exists a tiny cubical cavity in such a manner that two of...

Homework Statement Show that if p\in (1,\infty) the identity functions id:C^{0}_{1}[a,b]\longrightarrow C^{0}_{p}[a,b] and id:C^{0}_{p}[a,b]\longrightarrow C^{0}_{\infty}[a,b] are not continuous. Homework Equations C^{0}_{p}[a,b] is the space of continuous functions on the [a,b] with...

and then \lambda=\pm 1 necessarily and takes all values only if f\equiv 0. In fact zero function is symmetric and antisymmetric function at the same time.

Or that operator L is an involution, that's, it's its own inverse.

it tells me \lambda=1?