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.
17
votes
3answers
575 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 ...
16
votes
2answers
603 views
When is a notion of convergence induced by a topology?
I'm interested in sufficient conditions for a notion of sequential convergence to be induced by a topology. More precisely: Let $V$ be a vector space over $\mathbb{C}$ endowed with a notion $\tau$ of ...
11
votes
3answers
446 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 ...
10
votes
1answer
200 views
Is the standard structure of a topological vector space on reals unique?
The standard stucture of a topological vector space on reals is this given by the metric d(x,y)=|x-y| on the vector space $\mathbb{R},$ with the field of scalars $\mathbb R$ with standard topology.
I ...
10
votes
0answers
256 views
Differential forms on fuzzy manifolds
This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me.
Question
In the usual setting of open subsets of ...
9
votes
1answer
148 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 $X$ ...
8
votes
1answer
304 views
Isomorphisms of Fréchet Spaces
What is the proper notion of an isomorphism between Fréchet spaces? Obviously it should be a linear map. I'm just worried about the analytic structure. Should one be able to order the seminorms on ...
8
votes
1answer
256 views
Semi-Norms and the Definition of the Weak Topology
When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ...
8
votes
1answer
430 views
Hahn-Banach theorem: 2 versions
I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as:
Let $E$ be a vector space, $p: E \rightarrow \mathbb{R}$ be a sublinear function and $F$ be a subspace of ...
8
votes
2answers
550 views
If $A$ and $B$ are compact, then so is $A+B$.
This is an exercise in Chapter 1 from Rudin's Functional Analysis.
Prove the following:
Let $X$ be a topological vector space. If $A$ and $B$ are compact subsets of $X$, so is $A+B$.
My guess: ...
7
votes
3answers
397 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 ...
7
votes
2answers
694 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!
7
votes
1answer
168 views
Do the notions of weak and weak* convergence coincide for $\ell^1(\mathbb{N})$?
As my friends and I were studying for our real analysis final exam yesterday, we were playing with various examples and found ourselves asking this question:
The space $\ell^1(\mathbb{N})$ is the ...
7
votes
1answer
72 views
Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis
I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the ...
7
votes
1answer
116 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 ...
7
votes
1answer
218 views
Contractibility of convex set
Suppose that $\Omega$ is a convex open subset of an infinite dimensional vector space $E$ such that $\Omega$ is not contained in any finite dimensional subspace of $E$.
Let $Q_m\subset \Omega$ denote ...
6
votes
2answers
465 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 ...
6
votes
3answers
133 views
Can continuity of inverse be omitted from the definition of topological group?
According to Wikipedia, a topological group $G$ is a group and a topological space such that
$$ (x,y) \mapsto xy$$ and
$$ x \mapsto x^{-1}$$
are continuous. The second requirement follows from the ...
6
votes
1answer
205 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 ...
6
votes
1answer
304 views
The dual of a Fréchet space.
Let $\mathcal{F}$ be a Fréchet space (locally convex, Hausdorff, metrizable, with a family of seminorms ${\|~\|_n}$).
I've read that the dual $\mathcal{F}^*$ is never a Fréchet space, unless ...
6
votes
2answers
68 views
Alternate definition for boundedness in a TVS
Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$. A subset $B\subset X$ is defined to be bounded if
for any open neighborhood $N$ of $0$ there is a number $\lambda>0$
...
6
votes
0answers
111 views
Isomorphism between spaces of sections.
Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
5
votes
4answers
101 views
Books on locally convex topological vector spaces
My friend asked me for a good book about locally convex topological vector space. I'm not familar with this. Could you give me some good references on it?
5
votes
3answers
145 views
Is $C([0,1])$ a compact space?
Is $C([0,1])$ (I guesss with the max-norm) a compact space?
I have to know that because I want to apply Arzela Ascoli.
5
votes
2answers
93 views
Question about Topological Vector Spaces
Let $E$ be a Topological Vector Space and $U$ a bounded set of $E$ with $0\in U$, i.e. given any neighborhood $W$ of the origin, there exist $\alpha>0$ such that $\alpha U\subset W$. Is it true ...
5
votes
2answers
158 views
Local base of a topological vector space
I would like to prove that if $B$ is local base for a topological vector space $X$, then every member of $B$ contains the closure of some member of $B$.
I would appreciate if somebody can guide me ...
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? ...
5
votes
1answer
242 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 ...
5
votes
1answer
558 views
Convex functions and families of affine functions
I know that the supremum of a family of affine functions is convex. Just wondering if it is true (and if so how one proves) that the converse -- any $C^1$ convex function is the supremum of some ...
5
votes
1answer
88 views
Pseudonormable Product Spaces
I want to prove that a product $\prod_{i\in I}X_i$ of topological vector spaces is pseudonormable only if a finite number of the factor spaces are also pseudonormable and the rest have the trivial ...
5
votes
1answer
114 views
Bounded and compact sets in a subspace of $\mathbb R^{\mathbb N}$
Let
$$
X= \{u=(u_1, u_2, \ldots): u_n \ne 0 \text{ only for a finite number of terms}\}\subseteq\mathbb R^\mathbb N,
$$
with the topology inherited from $\mathbb R^\mathbb N$ (the "pointwise ...
5
votes
0answers
98 views
Evaluation map is not continuous always.
Let $E$ be a not normable locally convex space, define
$$F: E'\times E\to \mathbb R$$
$$(f,e)\to f(e)$$
I have to show that $F$ is not continuous when $E'\times E$ is given product topology.
I was ...
4
votes
4answers
125 views
Question on Topological vector space 1
I have numbered this question as (1) because I will be posting series of questions where I don't understand. I hope its allowed.
I want to prove the following :
If $X$ is a topological vector ...
4
votes
2answers
156 views
“The two notions of boundedness coincide for locally convex spaces”
From Wiki
The boundedness condition for linear operators on normed spaces can be
restated. An operator is bounded if it takes every bounded set to a
bounded set, and here is meant the more ...
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 ...
4
votes
2answers
222 views
Example of a topological vector space
I have the following question: give an example of a topological vector space $E$ with subspace $M$ and $N$, such that $E = M \oplus N$ algebraically, but not topologically (so $E \ncong M \sqcup N$). ...
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 ...
4
votes
2answers
184 views
Closed Bounded but not compact Subset of a Normed Vector Space
Consider $\ell^\infty $ the vector space of real bounded sequences endowed with the sup norm, that is
$||x|| = \sup_n |x_n|$ where $x = (x_n)_{n \in \Bbb N}$.
Prove that $B'(0,1) = \{x \in l^\infty ...
4
votes
1answer
108 views
Confused by proof in Rudin Functional Analysis, metrization of topological vector space with countable local base
I'm working through Rudin's Functional Analysis, and I am confused by a step in his proof for Theorem 1.24, which states that if X is a topological vector space with a countable local base, then there ...
4
votes
1answer
106 views
Connected components that are relatively open in $\sigma(T)$
Let $T$ be an bounded linear operator on a Banach space $X$. Suppose the spectrum of $T$, $\sigma(T)$ has infinitely many connected components, then $\sigma(T)$ must contain infinitely many ...
4
votes
1answer
164 views
Constructing a countable family of seminorms in a metrizable LCS.
Here's some context before my question.
Let $\mathbb{V}$ be a topological vector space, which is Hausdorff and such that its topology is generated by some arbitrary family of seminorms ...
4
votes
1answer
249 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 ...
4
votes
1answer
194 views
Uniqueness of the derivative in locally convex topological vector space
I need a hint of proof of uniqueness of the derivative in locally convex topological vector space (it's asserted in Lang's "Introduction to differentiable manifolds").
Define derivative of a function ...
4
votes
1answer
186 views
Why This Map is Closed?
Consider the following definition of closed maps, defined in the book Nonlinear Programming by Bazaraa et al.:
Let $X$ and $Y$ be nonempty closed sets in $\mathbb{R}^p$ and $\mathbb{R}^q$, ...
4
votes
1answer
146 views
Finest topology on a space of banach space operators
let $X$ be some Banach space. Let $L(X)$ be the set of continuous operators on $X$ into $X$.
Let $(\tau_i)_i$ be a set of topologies on $L(X)$ s.t. $L(X)$ is topological vector space (i.e. addition ...
4
votes
1answer
211 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 ...
4
votes
1answer
56 views
Openness of linear mapping 2
I quote a previously asked question :
Let $X$ be a topological vector space over the field $K$, where $K=\mathbb{R}$ or $K=\mathbb{C}$, and let $\mathbb\{f\colon X\rightarrow K^n\}$ ($n \in ...
4
votes
1answer
121 views
Existence of balanced neighborhoods in a topological vector space
I'm wondering about the following:
Let$\ X $ be a topological vector space. Then one could pick balanced neighborhoods$\ W $ and$\ U $ of$\ 0 $ such that
$\ \overline{U} + \overline{U} \subset W $, ...
4
votes
1answer
229 views
How to define the derivative of Radon measures
Let $M$ be the positive borel measures on a hausdorff topological space $X$, which are finite on compacts sets $--$ i.e. the real cone of radon measures.
I am given a definition of a derivative of ...
4
votes
0answers
104 views
Curiosities about the content of a rare book: Topological Vector Spaces by A. Grothendieck
The book is a celebrated and highly influential book by A. Grothendeck, which was published in 1954, in French and for various reasons, it has been out of print since 1973. I am very much interested ...
