3
votes
1answer
59 views

Sum of Neighborhoods of Zero

When do two neighborhoods of zero over a topological vector space add up as: $$aN+bN=(a+b)N\quad a,b\geq 0$$ I could imagine something like balanced might suffice... The problem is that I'd like to ...
1
vote
3answers
250 views

Counterexample for Mazur–Ulam theorem

We know, Mazur–Ulam theorem states that if $V$ and $W$ are normed spaces over $\mathbb{R}$ and the mapping $f\colon V\to W$ is a surjective isometry, then $f$ is affine. Can somebody say ...
0
votes
1answer
34 views

Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
2
votes
2answers
77 views

Show that no topological vector space is bounded.

I am studying the concept of topological vector spaces in Grubb's Distributions and Operators. A vector space $X$ (over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$) is called a topological vector ...
6
votes
2answers
159 views

Why do we need dual space [closed]

In functional analysis there are many places where dual space is mentioned, but I still don't understand the real power of that concept. Why do we need the dual space?
0
votes
2answers
79 views

Compactness in Infinite Dimensional Vector Spaces

Show that, in an infinite dimensional normed space $(V,\|\cdot\|)$, the closed ball of radius $2$ $$ B_2:=\{x\in V:\ \|x\|\leq2\} $$ is not compact. I suspect I am not understanding what is going ...
3
votes
1answer
98 views

Subspaces of a Topological Vector Spaces

I have a few questions about topological spaces which I am currently studying. First some definitions that I am using: Definition of subspace topology: Given a topological space $(X,\tau)$ and a ...
-2
votes
1answer
56 views

Are all metric space as a euclidean space?

I believe that all euclidean space is a metric space? But I need to know about inverse? I mean: are all metric space as a euclidean space? Is there any kind of metric space which is not euclidean ...
0
votes
1answer
272 views

closed subspace of normed vector space

Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.
1
vote
1answer
71 views

Set boundary preserved by an infinite union

Suppose I have a subset $U\subset\mathbb R^2$ and a real number $r>1$ with the following properties: $U$ is compact; $U\subset rU$ (self-similarity); $0\in U$; there exists an open set $H\subset ...
3
votes
1answer
588 views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
-1
votes
1answer
112 views

Practical implications of a vector space being a topological vector space

I have a space $V$ and I lately discovered that it's a topological vector space. What are the practical implications of that?
0
votes
0answers
68 views

Finding point distribution by eigen vectors

First of all I want to tell that my mathematics is poor, so I can’t use correct terms. Sorry for that. I have a point data set. This data represents some cylindrical objects surfaces (not exactly ...
0
votes
2answers
210 views

Vector Analysis & Linear Algebra

I'm given a positive number, a unit vector $u \in \mathbb{R} ^n $ and a sequence of vectors $ \{ b_k \} _{ k \geq 1} $ such that $|b_k - ku| \leq d $ for every $ k=1,2,...$. This obviously implies $ ...
0
votes
0answers
86 views

Can 2 parallel lines be discriminated as 'away', 'beside' with respect to 3rd parallel line?

I have nearly parallel several 3D line segments. some line segments locate (blue line) beside to a spefic line segment (black line) and some other (red line) locate away from that line segment. i want ...
0
votes
0answers
39 views

A subset that can be scaled to be the whole space, or that can contain a scaled version of the whole space

The following is from Mariano's comments on my earlier question In a topological vector space, why is the following true: if a neighborhood U of zero contains a scaled copy of the whole space, ...
1
vote
1answer
256 views

How to get a projected 3d line segment, lie on another 3d line parallel to that line segment.

I have a 3D line segment and another 3D position which locate slightly away from the line segment. I want to get the projected line segment (3D) which lies on imaginary 3D line which passes through ...
3
votes
0answers
100 views

Self-absorbing subsets in a vector space

From planetmath Let $V$ be a vector space over a field $F$ equipped with a non-discrete valuation $|\cdot|:F\to \mathbb{R}$ . Let $A$ and $B$ be two subsets of $V$. Then $A$ is said to absorb ...