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
2answers
99 views
Continuous linear functionals
Let L be a continuous linear functional on a metric linear space X. Prove: L(S) is a bounded set for any bounded subset S of X. The metric is translation invariant.
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 ...
2
votes
1answer
59 views
The weak topology of a product
Let $E$ and $F$ be normed vector spaces and let $E^{\sigma}$, resp. $F^{\sigma}$ be $E$, resp. $F$ with the weak topology associated with the elements of the duals $E^*$, resp. $F^*$. Then, for ...
2
votes
1answer
192 views
Closure of interior and interior of closure in a topological vector space
If $Y$ is a subset of topological vector space $X$ and is compact and convex show that $\overline{Y^\circ} = \overline{Y}$ and $\overline{Y}^\circ = Y^\circ$.
I tried this way but I am not sure:
...
1
vote
1answer
43 views
Questions regarding internal and interior points for a convex subset of a topological vector space
Suppose that $X$ is a topological vector space, with a convex subset $A$. How do we show that if the vector $u$ is in the interior of $A$, then $u$ is an internal point of $A$ and if the interior of ...
1
vote
1answer
232 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 ...
0
votes
1answer
56 views
Triangle inequality of a metric on a quotient space of a topological vector space
In "Functional Analysis" by Rudin, a metric $\rho$ on the quotient space $X/N$ of a topological vector space $X$ and a closed subspace $N$ is defined as follows:
For $x,y \in X$,
$$
\rho ...
10
votes
0answers
256 views
Differential forms on fuzzy manifolds
This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me.
Question
In the usual setting of open subsets of ...
7
votes
0answers
75 views
Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis
I am self-teaching myself the basics of functional analysis (e.g. topological vector spaces), and frankly I am starting to get a migraine sorting out/organizing in my head all of the ...
6
votes
0answers
111 views
Isomorphism between spaces of sections.
Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
5
votes
0answers
98 views
Evaluation map is not continuous always.
Let $E$ be a not normable locally convex space, define
$$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
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 ...
4
votes
0answers
161 views
Strong topology vs Natural topology
Let $X$ be a locally convex space and $\left< X, X^{\prime} \right>$ stands for the dual pair. The bidual of $X$ is denoted by $X^{\prime \prime}$ and this is a dual of $X^{\prime}$
with a ...
3
votes
0answers
184 views
About Henstock integrable vector-valued function
In what follows, $X$ is a Hausdorff locally convex topological vector space over the reals whose topology is generated by a family $P$ of all continuous seminorms on $X$. We consider the following ...
3
votes
0answers
89 views
Density of operators
I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by ...
3
votes
0answers
89 views
The converse of James's Theorem
The famous James theorem states that:
Theorem. Let $X$ be a (Hausdorff separated) locally convex space (LCS for short) with topological dual $X^*$ and let $B\subset X$ be weakly-closed. If $X$ is ...
3
votes
0answers
114 views
Locally convex space characterization in terms of duality
Let $(X,\tau)$ be a topological vector space (TVS for short). Denote by $X^*$ the topological dual of $(X,\tau)$.
If there exists a locally convex topology $\mu$ on $X$ compatible with the duality ...
3
votes
0answers
82 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 ...
3
votes
0answers
232 views
Understanding examples - metric spaces, Minkowski functionals and topologies
I'm teaching myself a course on functional analysis but having trouble understanding the notes I've been using. I was hoping I could write out a section of the content and you might be able to help me ...
2
votes
0answers
61 views
Which are nontrivial examples of analytical functions on Frechet spaces?
Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued ...
2
votes
0answers
35 views
Density problem
$U$ is any open set of $\mathbb{R}$. We known that $C_0^\infty(U)$ is dense in $C^k(U)$. But what about, say $C_0^\infty((0,1))$ in $C^k([0,1])$?
2
votes
0answers
65 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 ...
2
votes
0answers
184 views
Translation invariant metric
Under what conditions can a metric vector space be given an equivalent metric that is translation invariant?
I was wondering if the probability measures on real line can be embedded in vector space ...
2
votes
0answers
45 views
Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?
Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation:
Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
2
votes
0answers
140 views
Definition of a topological module
A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
2
votes
0answers
70 views
Consequence of metrizability proof - disregard, the question is an error
In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if $X$ is a separable Banach space, then the (closed) unit ball, $B_{X^{*}}$ of ...
2
votes
0answers
41 views
When is a subset of {0,1} valued borel functions on a standard borel space (polish space) complete (see *) under the pointwise convergence topology?
*In other words, what restrictions on a family F of {0,1} valued borel functions will tell us that the pointwise limit of any net in F is borel. I feel like there must be lots known about this but I ...
1
vote
0answers
36 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 ...
1
vote
0answers
39 views
Two different opinions on whether a topological vector space is a uniform space
Van de Vel's Theory on Convexity Structures says a TVS is uniform iff it is locally convex:
3.10.1. Proposition. Let $X$ be a topological vector space, equipped with the
standard convexity and ...
1
vote
0answers
46 views
Determining Similarity of Unit Vectors
I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$.
The piecewise continuity requirement ...
1
vote
0answers
51 views
Finite Dimensional TVS
Let $E, F$ topological vector spaces, $E$ normable and $T: E \longrightarrow F$ linear, compact and surjective. Show that $\mbox{dim}(F)< \infty$.
1
vote
0answers
60 views
Inductive limits
Let $E_n$ be a family of Banach spaces. Under which conditions imposed on $(E_n)$ can we represent the $\ell_\infty$-sum $(\bigoplus_{n\in \mathbb{N}} E_n)_{\ell_\infty}$ as a complemented subspace of ...
1
vote
0answers
133 views
Hairy ball theorem: references to applications
I'm looking for references to applications of the Hairy ball theorem.
I already visited wikipedia and cited references, but I need a little more explanation in both meteorology and applications in ...
1
vote
0answers
109 views
Are topological vector spaces completely regular?
Every uniformizable space is a completely regular topological
space.
topological vector spaces are uniform spaces.
every Hausdorff topological vector space is completely regular.
From ...
1
vote
0answers
81 views
Are nuclear Montel spaces projective?
The well known fact about $\ell^1$ says that the Schauder basis of $\ell^1(I)$ behaves more-less like a Hamel basis, namely if $X$ is any Banach space and $\mathcal{E}=(e_i)_{i\in I}$ is a basis for ...
0
votes
0answers
28 views
Is this question stated wrong?
I'm trying to check whether this question might be worded wrong, and here it is:
Show that if $A$ is a convex subset of a topological vector space $X$, $u \in A^o$ (the interior of $A$), $v \in ...
0
votes
0answers
27 views
Proving the density of a set into another set
I am dealing with the proofs for some density theorems in the context of Functional Analysis. I know that one can prove that $\bar{X} = Y$ by means of constructing a sequence $\left\{f_n \right\}_n ...
0
votes
0answers
61 views
Is a closed set in a TVS over $\mathbb{R}$ convex?
From Theory of Convex Structures by M. L. J. Van De Vel, on a set $X$, a topology and a convexity structure are said to be compatible, if the convexity structure is generated by the closed sets. The ...
0
votes
0answers
41 views
Linear Application that is open in a TVS
Let $T: E \to F$ be a linear map between topological vector spaces $E$, $F$. If for each nonempty open set $G$, the interior of $T(G)$ is non-empty, then, $T$ is open.
Proof:
$$\mathrm{Int}(T(G))= ...
0
votes
0answers
48 views
functions from the sphere
Can we assign in a continuous manner to each point of the sphere $S^2$
a two point subset of S^2?
I think this would contradict in some way "The Poincare theorem"
Thanks
0
votes
0answers
62 views
Topological vector spaces question regarding dual spaces
there's a question that I've be puzzling over for a little while now and I haven't made much progress, so I thought I'd better ask for some help. It is as follows:
Suppose that $\langle E,F\rangle$ ...
0
votes
0answers
51 views
When is the orbit of a vector a minimal sequence? When does an operator have a minimal orbit vector?
Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors.
For ...
0
votes
0answers
51 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
0answers
78 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
31 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,
...
