# locally

1. ### Locally convex set

Considering a constrained nonlinear programming (NLP) problem min \quad f({\bf x}) \quad {\bf x}\in \mathbb{R}^{n} s.t. \quad g_{i}({\bf x})\leq 0 \quad i=1,2,...,N \quad\quad\quad\quad h_{j}({\bf x})=0 \quad j=1,2,...,M Where g_{i}({\bf x}) and h_{j}({\bf x}) is twice continuously...
2. ### Locally Lipschitz function is bounded

Hoi, I'm trying to prove that in a banach space: If f is locally lipschitz, then it maps bounded sets onto bounded sets. Sounds fair right? I started an argument as follows, but I couldn't finish it... Choose x_1\in B, and let U_1:=U(x_1) be the largest neighborhood in B containing x_1, such...
3. ### Locally compact Hausdorff space

In a locally compact hausdorff space X for every x in X there exists a compact set that contains an open set V s.t. x is in V. I have to prove this is equivalent to for every open set U s.t. x is in U, there exists a open set V s.t x is in V and closure of V is compact and contained in U. I know...
4. ### locally compact and proper functions

Hello, i try to solve following exercise: Let f:X-> Y be an continuous immersion. X a Hausdorff-space and Y locally compact, i.e. Y is a Hausdorff-space with the property that every point y \in Y has a nbh. U with closure(U) compact. Show that f is proper iff f^(-1) (K) \subset X is compact...
5. ### locally constant

Hello, I want to show the following : Let U be a open subset of IR^2. If f is constant on each connected component, then f:U->\IR is locally constant. My Idea was this: Let x be a arbitrary point of U. We have to show that there is a (open) nbh. V of x, s.t. f|V is constant. We know x is an...
6. ### Locally Path Connected

Are the rational numbers Q, with the relative topology as a subset of the real numbers, locally compact ? why?
7. ### locally nilpotent operator.

Let T:V \to V nilpotent linear transformation over some field, let assume that dimV=n. Prove that T nilpotent if and only if the characteristic polynomial of T is x^n Thank you!
8. ### Bases in Locally Euclidean spaces

Hey everyone, I think I am making the following WAY too difficult. Does anyone have insight into A) whether the following is even correct or B) an easier way to do it (as I am sure there is one). Problem: Let \mathfrak{M} be a locally Euclidean space of dimension n. Call an open subset E of...
9. ### Prove the solution is locally the sum of...

Prove that any solution of the equation is locally the sum of a constant and the distance to some curve.
10. ### Locally Exact Differential....

A differential pdx+qdy(in \mathbb {C})is said to be a locally exact differential in a region \Omegaif it is an exact differential in some neighbourhood of each x\in\Omega. Show that every rectangle with sides parallel to axes R\subset\Omega satifies \int_{\partial R}pdx+qdy=0if it is a locally...
11. ### solving specific equations locally?

can these equations be solved locally near u=1, v=2 for c1 functions x=x(u,v) , y=y(u,v) with x(1,2) =1 , y(1,2)= -1 ... these are the equations: u^2xy^2 + uvx+y = 2 vx^2 + u^2y^2 + uv =5 --- If so, find the partial of x in respect to u at (1,2) . ? Thanks! :)

13. ### locally connected (topology)

Munkres 3.25 exercise 8 says: Let p:{X}\rightarrow{Y} be a quotient map. Show that if X is locally connected then Y is locally connected. To do this we consider a component C of the open set U of Y. We should show that p^{-1}(C) is a union of components of p^{-1}(U). Anybody have any...
14. ### locally measurable functions

Is there any definition for locally measurable functions similar to locally continuity ?
15. ### Is every strictly locally monotone sequence strictly monotone?

Sequence a_n, n=1,2,3... is strictly locally monotone if for every integer k>1 either a_{k-1}<a_k < a_{k+1} OR a_{k-1}> a_k > a_{k+1}. Prove every strictly locally monotone sequence is strictly monotone or give counterexample.