1
vote
1answer
28 views

Existence of right inverse.

We know that a surjective continuous linear map $ T : X \to Y$ has a right inverse iff $ \ker(T)$ is complemented. Here $X$ and $Y$ are Banach spaces. Is this result true for locally convex ...
5
votes
0answers
102 views

Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
0
votes
0answers
23 views

Linearly compact subspace is closed

Let $V$ be a vector space with a linear topology, meaning open linear subspaces form a basis of open neighborhoods of $0$. Recall that a subspace $W$ is called linearly compact if for any open ...
1
vote
0answers
34 views

Direct limit, density and norm

Let $E$ be a Banach space, $A_n$ be an increasing sequence of finite dimensional subspaces of $E$, $B_n$ be an increasing sequence of subspaces of $A_n$ and let $C_n = A_n/B_n$. Assume that the ...
0
votes
2answers
20 views

if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member of $K$. How is this possible?

Let $(V,K)$ and $u,v$ is a member of $V$. Suppose that $M$ is a subset of $V$ is a subspace of $V$ with basis $B_m=\{m_1,...,m_r\}$ with $r$ less than and equal to $n$. Let $H$ be a subspace spanned ...
1
vote
0answers
47 views

A separation lemma in a real vector space

A lattice $N$ is a free $\mathbb{Z}$-module of finite rank. Let $V$ be the real vector space $N\otimes_\mathbb{Z} \mathbb{R}.$ A cone is a set $\sigma = \{ r_1 v_1 + \ldots + r_k v_k \in V : r_i\geq 0 ...
4
votes
2answers
311 views

Topology of the space of hermitian positive definite matrices

Let $\mathcal{H}_n \mathbb{C}$ be the set of hermitian $n \times n$ complex matrices. This set carries the structure of a vector space over $\mathbb{R}$ under usual addition. It also inherits the ...
1
vote
0answers
48 views

Show that the norm of the derivitive of a $C^1$ function over a vector space is non-negative, homogeneous and satisfies the triangle ineq

For $f$ in $C^1[a,b]$, define $p(f)= \parallel f'\parallel _{\infty}$. Show that $p$ is non-negative, homogeneous, and satisfies the triangle inequality. Why is it not a norm? -I can easily show the ...
3
votes
2answers
202 views

Is there a dumbed down version of the open mapping theorem for finite dimensional real vector spaces?

I would like to understand why any surjective linear transformation between finite dimensional topological real vector spaces, each with the natural topology, is in fact an open map. Reading around, ...
2
votes
3answers
170 views

Are vector subspaces of $\mathbb{R}^n$ always closed?

Suppose $S$ is any proper vector subspace of $\mathbb{R}^n$. Is $S$ a closed set in the usual topology on $\mathbb{R}^n$? Geometrically, I think it is clear that $S$ must be closed in $\mathbb{R}^n$ ...
3
votes
0answers
103 views

Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
0
votes
2answers
215 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 $ ...
6
votes
1answer
552 views

Generic topology on a vector space?

For a (possibly infinite-dimensional) vector space $V$, I thought about the following topology $\tau$: Let $O \in \tau$ if every $x \in O$ has the property that for every $v \in V$, there is an ...
3
votes
2answers
474 views

projection operators on topological vector spaces

Suppose $A \in \mathbb{R}^{m\times n}$. Then there exists a projection matrix $P$ onto the range of $A$. In other words, there exists a matrix $P \in \mathbb{R}^{m\times m}$ such that $P^2=P$, and ...
11
votes
3answers
577 views

Topology on the general linear group of a topological vector space

Let $K$ be a topological field. Let $V$ be a topological vector space over $K$ (if it makes things convenient, you may assume it is finite dimensional). Naive Question: Is there a canonical way of ...