The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.
4
votes
1answer
1k views
“Every linear mapping on a finite dimensional space is continuous”
From Wiki
Every linear function on a finite-dimensional space is continuous.
I was wondering what the domain and codomain of such linear function are?
Are they any two topological vector ...
0
votes
1answer
218 views
Weak *-topology of $X^*$ is metrizable if and only if …
Let $X$ be a topological vector space on which $X^*$ separates points. Prove that "the weak *-topology of $X^*$ is metrizable if and only if $X$ has a finite or countable Hamel basis"?
(A set $\beta$ ...
3
votes
1answer
284 views
$\omega$ - space of all sequences with Fréchet metric
I'm working on to prove the following:
Show that the convergence in the space $\omega$ (space of all sequences with respect to the Fréchet metric) is the coordinate convergence.
Any hint is ...
7
votes
3answers
401 views
Do continuous linear functions between Banach spaces extend?
Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
5
votes
1answer
79 views
local convexity of $L_p$ spaces
wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm
they are not locally convex, since the only convex neighborhood of zero is the whole space
Why is this so? ...
3
votes
1answer
304 views
Meaning of “a mapping preserves structures/properties”
Sometimes I see something like "a mapping preserves the structures
of its domain and of its codomain". From Wiki about morphisms in category theory:
a morphism is an abstraction derived from ...
7
votes
2answers
700 views
Dual of a dual cone
Any hint on how to prove the following please:
Let $K$ be a convex cone, and $K^*$ its dual cone. Prove that $K^{**}$ is the closure of $K$.
Thanks!
3
votes
3answers
989 views
The kernel of a continuous linear operator is a closed subspace?
If $V$ and $W$ are topological vector spaces (and $W$ is finite-dimensional) then a linear operator $L\colon V\to W$ is continuous if and only if the kernel of $L$ is a closed subspace of $V$.
...
2
votes
1answer
103 views
What is the topological dual of a dual space with the weak* topology?
I'm trying to understand a claim I heard in class. To be concrete, suppose $X$ is a compact, hausdorff topological space, and let $C(X)$ be the space of continuous functions on $X$ with the supremum ...
7
votes
1answer
117 views
Rotation of $\mathbb{R}^3$ by using quaternion
Express the rotation of $\mathbb{R}^3$ by $\frac{\pi}{3}$ about the $x=y=z$ axis by using quaternions and identifying $\mathbb{R}^3$ with $(i,j,k)$-space.
Thoughts:
From my point of view, every ...
4
votes
1answer
215 views
Is any Banach space a dual space?
Let $X$ be a Banach space. Is there always a normed vector space $Y$ such that $X$ and $Y^*$ are isometric or isomorphic as topological vector spaces (that is, there exists a linear homeomorphism ...
2
votes
1answer
125 views
An explicit construction for a “doubly weak” topology
Let $(X,s)$ be a topological vector space over $\mathbb{F}$ with linear topology $s$, which we will henceforth refer to as the strong topology. Then, as usual we can construct the continuous dual ...
1
vote
1answer
44 views
About a weak topology on TVS
Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
17
votes
3answers
578 views
When do weak and original topology coincide?
Let $X$ be a topological vector space with topology $T$.
When is the weak topology on $X$ the same as $T$? Of course we always have $T_{weak} \subset T$ by definition but when is $T \subset ...
9
votes
1answer
150 views
If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?
Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive?
To prove the contrapositive, it will suffice to assume that $X$ ...
5
votes
1answer
243 views
Sequential and topological duals of test function spaces
Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test ...
1
vote
2answers
116 views
Left topological zero-divisors in Banach algebras.
Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by
$$
\forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|,
$$
where $ ...
6
votes
2answers
478 views
Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?
Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ?
(Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
4
votes
1answer
257 views
Finding the topological complement of a finite dimensional subspace
I know that for any finite dimensional subspace $F$ of a banach space $X$, there is always a closed subspace $W$ such that $X=W\oplus F$, that is, any finite dimensional subspace of a banach space is ...
2
votes
1answer
195 views
Definition of boundedness in topological vector spaces
From Wikipedia:
Given a topological vector space $(X,τ)$ over a field $F$, $S$ is
called bounded if for every neighborhood $N$ of the zero vector there
exists a scalar $α$ so that $$
S ...
2
votes
1answer
124 views
Simple Continuity Question concerning Vector Bundles
I'm currently reading M. Atiyah's "K-Theory," and I'm a bit confused about the following part. Let $E = X \times V$ and $F = X\times W$, and let $\phi: E\to F$ be a vector bundle homomorphism. Why ...
11
votes
3answers
449 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 ...
4
votes
2answers
215 views
If you know the convergent sequences, how do you know the open sets?
I have a homework problem which I feel should be simple but is actually surprisingly tricky. This is why I love math sometimes....
Let $X$ be a normed linear space. Suppose $\|\cdot\|_1$ and ...
3
votes
1answer
177 views
Extreme boundary of a compact, convex, metrizable set is $G_\delta$
Let $X$ be a topological vector space (no assumptions about local convexity are made in the question, though I am worried they might be required). Suppose $K\subset X$ is a compact, convex, metrizable ...
2
votes
2answers
163 views
Openness of linear mapping
Let $X$ be a topological vector space over the field $K$, where $K=\mathbb R$ or $K= \mathbb C$, and let $f:X\rightarrow K^n$ ($n \in \mathbb N$)
be a linear and surjective functional. How to prove ...
1
vote
1answer
34 views
About a Weak Topology on TVS(part 2)
Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
1
vote
1answer
270 views
Continuous linear mapping and bounded subsets
Continuous linear mappings between topological vector spaces preserve
boundedness.
I was wondering if it means that the inverse image of a bounded
subset under a continuous linear mapping is ...
0
votes
2answers
137 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 $ ...
