5
votes
1answer
42 views

Is this condition sufficient to determine the linear space is of finite dimension?

From the Banach theory we knew that: 1) A linear space(a vector space endowed with its vector topology) $X$ of finite dimesion $dimX=n$ has the following property: If ${\left\| \bullet \right\|_1}$ ...
0
votes
1answer
41 views

Proof about non-compact sets and unbounded functions

Let $A \subset \mathbb{R}^n$ be a non-compact subset. Show that there exists a continuous unbounded function on $A$. I have split this into two parts. Either: $A$ is unbounded but closed, or $A$ ...
1
vote
0answers
37 views

understanding of separable space

i would like to understand correctly what does mean separable space,from wikipedia i am reading that In mathematics a topological space is called separable if it contains a countable, dense subset; ...
3
votes
3answers
75 views

For closed sets, is $\text{cl}(A+B)=\text{cl}(\text{cl}(A)+\text{cl}(B))$?

Let $A$ and $B$ be nonempty subsets of $\mathbb{R}^n$, then is $\text{cl}(A+B)$ equal to $\text{cl}(\text{cl}(A)+\text{cl}(B))$? If that is true, then how to prove it? If they are not equal, then ...
0
votes
1answer
63 views

Show that every finite-dimensional topological vector subspace is closed.

Let $X$ be a normed topological vector space. Show the following: (i) If $0\neq v \in X$, then $\{\alpha v:\alpha\in \mathbb{R}\}$ is closed. (ii) If $Y$ is a closed vector subspace of $X$ and $w\in ...
3
votes
1answer
84 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 ...
1
vote
1answer
27 views

Can we construct two sets and functions for the given conditions?

Can we construct two sets $A$, $B$ and two invertible functions (one to one) $f_A \in \mathbb{R}^n$, $f_B\in \mathbb{R}^n$ such that the following conditions are satisfied? The conditions are ...
0
votes
1answer
126 views

Is the set closed, open, or neither?

Consider $C[0,1]$, the normed linear space of all real-valued continuous functions within the given interval. The norm endowed on this space is $\|f\|_{\infty} = \sup_{x \in [0,1]} |f(x)|$. Consider ...
1
vote
1answer
60 views

The subspace of complex sequences $V_{\sigma}$ is closed under what topology?

Let $C^\mathbb{N}$ be the vector space of complex sequences. Let $V_{\sigma}$ be the vector subspace of $C^\mathbb{N}$ such that $(a_i) \in V_{\sigma}$ iff $\sum_{i=1}^{\infty} a_{\sigma(i)} \text{ ...
1
vote
1answer
28 views

The idea of the transverse to a vector field

I have a quick question. I am independently reading a book on three dimensional geometry and topology. One line has been stumping me. Here is the following paragraph I do not understand: "Let $X$ be a ...
0
votes
0answers
19 views

isolated zero z of X on a star shaped polygon

Consider a polygon that is star shaped with respect to the isolated zero $z$ of $X$. I want to show that the boundary of the polygon can be made transverse to $X$ by jiggling vertices only in the ...
1
vote
0answers
50 views

Can you construct a coutable local base in the space of continuous functions?

Let $(C,\tau)$ be the topological vector space of all complex continuous functions on $[0,1]$ with seminorms $p_x(f)=|f(x)|$, $x\in [0,1]$. We have known $(C,\tau)$ is not metrizable,but how could I ...
0
votes
3answers
159 views

Closure point and closure set

Question: If $P\subseteq \mathbb{R}^n$, how to show that $\overline{\overline{P}}=\overline{P}$, i.e. the closure of $\overline{P}$ equals the closure of $P$. I know that in a vector space with a ...
1
vote
1answer
40 views

The algebraic possibilities of the (topological) procedure of the compactification of a space

If $X$ is locally compact $K$-vector space, then $X\cup \{\infty\}$ is via the Alexandroff-compactification a compact space. But this purely topological procedure tells me nothing about the algebraic ...
1
vote
1answer
172 views

Direct (Inductive) limit of (locally convex) TVEs and universal property

This is not really a question, I'd just like to discuss a little about universal properties (more specifically, the direct limit) in TVEs. I'm trying to work with universal properties in Topological ...
2
votes
1answer
64 views

$[T]^{\beta}_{\beta} = \begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix}$ provided $T \circ T = T$ [closed]

Let $V$ be a finite-dimensional vector space and let $T:V \rightarrow V$ be a linear map such that $T \circ T = T$. How should one prove that there is a basis $\beta$ of $V$ such that \begin{eqnarray} ...
0
votes
1answer
44 views

Codimensionality: On Cardinality of Linear Equations

How does the codimension of a subspace give the number of linear equations needed to define the subspace?
3
votes
1answer
114 views

Dense subspaces

How does one go about proving the following statements? (a) $\operatorname{Lip}[a,b]$ functions are dense in absolutely continuous functions on $[a,b]$ under the variation norm - (Another doubt: what ...
2
votes
2answers
91 views

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$?

How to show $X=\{A\in\mathcal{L}(\mathbb R^m, \mathbb R^n); \textrm{Ker}(A)=\{0\}\}$ is open in $\mathcal{L}(\mathbb R^m, \mathbb R^n)$? Here $\mathcal{L}(\mathbb R^m, \mathbb R^n)$ is the set of ...
2
votes
5answers
179 views

How to show that the vector subspaces of $\mathbb{R}^{n}$ are closed in $\mathbb{R}^{n}$?

The vector subspaces of $\mathbb{R}^{n}$ are closed in $\mathbb{R}^{n}$. How to show this?
3
votes
2answers
61 views

Proving that certain subspace of $\ell_1$ is non closed

I need to prove that $$L= \left\{(x_i) \in\ell_1 : \sum_{i=1}^\infty ix_i= 0\right\}$$ is non-closed in $\ell_1$. I can't really think of sequences of sequences that are in this subspace, much less ...
2
votes
1answer
150 views

Prove that a given subspace of $C[-1,1]$ with $L^2$ norm is closed

Let $H= C[-1,1]$ with $L^2$ norm and consider $G=\{f \in H \mid f(1) = 0\}$. Show that $G$ is a closed subspace of $H$. I've been trying to prove this for a while but i can't establish that given ...
4
votes
1answer
210 views

Looking for proof that an open set in vector space contains the sum of two open sets.

Problem: To show that, in a topological vector space, for a given neighborhood of zero $W$, there exist two neighborhoods of zero, $V_1$, $V_2$, whose sum is contained in the first neighborhood, ...
0
votes
4answers
512 views

Compact set - prove that supremum is actually maximum

There is a compact set $K$ in $\mathbb{R}^n$. The diameter of this set is defined as follows: $D = \sup\limits_{x,\, y\, \in K}\|x-y\|$. I need to prove there are two vectors $a,b$ in $K$ such ...
4
votes
1answer
193 views

Smallest/Minimal bases of a topological space

The smallest possible cardinality of a base is called the weight of the topological space. I was wondering if all minimal bases have the same cardinality, and if every base contains a subset whose ...
0
votes
1answer
122 views

Topology Vector Spaces

I am new to topology & vector spaces with rudimentary knowledge. Could any one suggest me a comprehensive book on topology with vector spaces ? I find topology & vector spaces - mathematical ...
1
vote
1answer
67 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 ...
2
votes
1answer
339 views

If $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded.

As in the title, if $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded. The $\sigma(X,X^{\ast})$ topology is the weakest topology that makes linear ...
2
votes
0answers
51 views

Continuity of linear form

Let $E=\mathbb{R}[X]$ We define $N:\, P \to \sum_{n=0}^{\infty} { |P^{(n)}(n)|}$ ($P^{(n)}$ being the $n$-th derivative) , it is not hard to prove that $N$ is a norm on $E$. Help me to study the ...
0
votes
0answers
137 views

Finite dimensional normed space

I would like to find an elementary proof of the following theorem Let $E$ be a normed space. Then the following statements are equivalent: (a) E is finite dimensional. (b) Every linear functional ...
0
votes
1answer
117 views

Topological vector space generated by weak topology

Q1. Let $(X, \|.\|)$ be a real Banach space and $\tau$ is the weak topology on $X$. I would like to ask whether $(X,\tau)$ is a topological vector space? Q2. Let $(X, \|.\|)$ be a real Banach space ...
4
votes
5answers
707 views

Subspaces of Hilbert Spaces of finite dimension

Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ ...
1
vote
1answer
197 views

Homotopy equivalence of Unit Balls [duplicate]

Possible Duplicate: $S^n \backslash S^m $ homotopy equivalent to $ S^{n-m-1} $ I'm trying to show that, if we embed $S^m$ in $S^n$ as the subspace $\{ (x_1,x_2, \ldots, x_{m+1}, 0, 0, ...