Questions tagged [topological-vector-spaces]
A topological vector space is a vector space $V$ over a topological field $\mathbb{K}$ (typically $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), together with a topology on $V$ such that vector addition and scalar multiplication are both continuous. Hilbert spaces and Banach spaces are examples of topological vector spaces.
347 questions
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Understanding bounded subsets and open subsets in strict inductive limits of topological vector spaces
Let $\{H_i\}_{i\in I}$ be a directed system of Hilbert spaces over a poset $I$ such that $H_i\subset H_j$ whenever $i\leq j$. We consider the direct limit $H=\bigcup_{i\in I} H_i$ and equip it with ...
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Understanding quasibarrelled-ness in $C(X)$
Definition: A locally convex topological vector space is called quasi-barrelled if every bornivorous barrel is a neighborhood of 0.
A bornivorous set in a tvs is a subset which absorbs all bounded ...
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What locally convex manifolds have $\Delta$-generated topology?
Recall that $\Delta$-generated topological spaces is a topological space whose topology is determined by the continuous maps from standard topological simplices, which is a variant on talking about ...
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Balls which are bounded weak neighborhoods of $0$
Let $E$ be a Banach space. Let $B$ be the closed unit ball of $E$ endowed with the restriction of the weak topology of $E$. For $e\in E$, $r\in\mathbb{R}$ let $B(e,r)$ be the closed ball of radius $r$ ...
- 5,693
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Local base for direct sum of locally convex spaces
In Schaefer "Topological vector spaces" on page 55, he defines the direct sum of a family of locally convex spaces. If $(E_\alpha)$ is a family of locally convex spaces, their direct sum is ...
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When the sum of a homeomorphism and a continuous map is a homeomorphism between TVSs
Let $X,Y$ be topological vector spaces of positive dimension and $f,\iota:X\to Y$ be continuous maps such that $\iota$ is an embedding. Is there is a simply-verifiable criterion to test weather $f+\...
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Is the compact-open topology Fréchet–Urysohn?
Let $(X, d)$ be a complete and separable metric space. Let $\mathcal C (X)$ be the space of all continuous real-valued functions on $X$. We endow $\mathcal C (X)$ with the topology induced by uniform ...
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When is the quotient of an LF space by a closed subspace LF (Reference request)
I know that the quotient of a Fréchet space by a closed subspace is Fréchet
(see for example https://math.stackexchange.com/questions/4735928/is-the-fr%C3%A9chet-quotient-space-given-by-the-induced-...
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(op)Cartesian morphisms of Vector Bundles
Considering the forgetful functor $U \colon \mbox{VBun} \to \mbox{Top}$ from the category of topological vector bundles into the category of topological spaces. Is this a fibration and(or) opfibration?...
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When do bilinear forms on continuous functions correspond to Radon measures?
Let $\Omega \subset \mathbb{R}^N$ be an open set and denote by $C_c^0(\Omega)$ the set of compactly supported continuous functions on $\Omega$. We assume that $B : C_c^0(\Omega) \times C_c^0(\Omega) \...
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On the boundary of Fredholm operators in Fréchet spaces
Crossposted from Math Stack Exchange
It is well known that, given a Banach space $X$, the boundary of the set of Fredholm operators does not contain any semi-Fredholm operators (i.e. operators with ...
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Intersection of ball and a sphere is connected in the normed vector space
Studying functional analysis, I have accidentally formulated the following statement:
Claim. Suppose that $(X, \Vert \cdot \Vert)$ is a normed vector space (over the field of real or complex numbers) ...
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Rearrangement of finitely many elements in Schatten $p$-classes
Let $a=(a_1,a_2,\dotsc,a_n)$. Assume that $\|a\|_2=1$.
We can find a rearrangement of $a$, called $a'=(a_1',a_2',\dotsc,a_n')$, such that
$$\|\sum_{i=1}^k a_ i'\|_2 \le k^{1/2} n^{-1/2} =\|(\...
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Rearrangement in finite-dimensional $l_2$ and Schatten $p$-classes
Let $a=(a_1,a_2,\dotsc,a_n)$. Assume that $\|a\|_2=1$.
Can we find a rearrangement of $a$, called $a'=(a_1',a_2',\dotsc,a_n')$, such that
$$\|\sum_{i=1}^k a_ i'\|_2 \le k^{1/2} n^{-1/2} =\|(\...
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Associativity of projective tensor product
Let $X$ and $Y$ be Hausdorff topological vector spaces. The projective topology is the strongest topology on algebraic tensor space $X\otimes Y$ such that the canonical map $X\times Y\to X\otimes Y$ ...
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Inverse limits and formal power series: Does $\varprojlim V_i[[\hbar]] = V[[\hbar]]$ imply $\varprojlim V_i = V$?
Let $K = \mathbb{C}[[\hbar]] $, and consider an inverse system of surjective $ K $-linear maps
$$ f_{ij} \colon V_j[[\hbar]] \twoheadrightarrow V_i[[\hbar]] $$
for $ i, j \in \mathbb{Z}_{\geq 0} $ ...
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Example of topological vector space such that its closed subspaces are complemented
It is known that any Banach space that is such that any closed subspace is complemented is isomorphic to a Hilbert space.
My question : is there a Frechet space that is non-isomorphic to a Banach ...
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Is there any work on point-free topological vector spaces?
Is there any substantial work on topological vector spaces in the point-free setting? It seems like this might be able to clarify some of the traditional pathologies of the subject.
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The space of bounded smooth functions with rapidly decaying derivatives
Let $f : \mathbb{R}^n \to \mathbb{C}$ be a bounded smooth function such that all of its partial derivatives are rapidly decaying. That is, for any nonzero $n$-dimensional multi-index $\alpha$, $D^\...
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Operators with closed complemented ranges and surjective operators
My questions are in the spirite of Bounded operators with closed complemented ranges.
Let $F$ be a Fréchet space and $A:F\rightarrow F$ a continuous operator such that $A\left( F\right) $ is closed ...
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Sheaves of modules over a topologically ringed space
Topologically ringed spaces turn up a few places in the literature (most notably in EGA 1's construction of formal schemes), but I can't find much general information about them (e.g. the nLab page is ...
- 1,474
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Literature on Fréchet quasi-coherent sheaves
I've heard/read it said many times that "the good notion of quasi-coherent sheaf for complex manifolds is that of a Fréchet quasi-coherent sheaf", and the standard citation to which I've ...
- 1,474
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There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions
Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$.
Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
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Continuum-distanced complete, ultrametric space
Our professor asked us to find a complete metric space where the intersection of nested closed balls can be empty.
The following space is such an example, and I would like to learn more on it (since ...
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Analytic functions and Hyperfunction as TVS
I have several related questions on Analytic functions and Hyperfunction as topological vector spaces (I am mainly interested in questions 4,6,10):
For an open set $U\subset \mathbb C^n$ we can ...
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Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product
Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing).
For any pair of ...
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Unique Hausdorff topology on trivial vector bundle?
Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
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Is any submetrizable linear topology linearly submetrizable?
Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$.
Is ...
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Topology on a module over a topological ring
The questions
Let $R$ be a topological ring, and let $M$ (with no topology) be an $R$-module. Does $M$ somehow "inherit" a topology from the action of $R$?
Here's a proposal for such a ...
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Boundedness of 2 times the unit ball
Suppose that $X$ is a topological vector space where the topology is given by a metric $d$ on $X$. Assuming that the unit ball
$$
B(0, 1) := \{x \in X : d(0, x) < 1\} \neq X,
$$
is it necessarily ...
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A sensible topology on the space of continuous linear maps between Fréchet spaces
Let $V_1$ and $V_2$ be Fréchet spaces. Let $\{ \lVert \cdot \rVert_{1,n} \}_{n \in \mathbb{N}}$ be a family of seminorms for $V_1$ and similarly $\{ \lVert \cdot \rVert_{2,n} \}_{n \in \mathbb{N}}$ ...
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Heine-Borel property for (probability) measures on $\mathcal{S}'$?
For Euclidean space $\mathbb{R}^n$, it is a well-known fact that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a Montel space. Moreover, the space of tempered distributions $\mathcal{S}'(\mathbb{R}...
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Density of Schwartz distributions in the space of distribution
Let $\mathscr S(\mathbf R^3)$ and $\mathscr D(\mathbf R^3 )$ be the space of Schwartz function and test function respectively, $\mathscr S'(\mathbf R^3)$ and $\mathscr D'(\mathbf R^3)$ be their duals....
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Is compact-open topology stable with respect to injective limits?
Let $X$ be a locally convex space, and $\{Y_i;\ i\in I\}$ a covariant system of locally convex spaces over a partially ordered set $I$, i.e. a system of linear continuous mappings is given $\sigma^j_i:...
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Colimits of locally convex spaces in the categories of topological vector spaces vs locally convex spaces
Let $S$ be a set and let $V_s$ be a family of locally convex topological vector spaces (LCSs) indexed by $s \in S$. Let $V$ be a vector space (without topology) and let $T_s:V_s \to V$ be a family of ...
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Are dualizable topological vector spaces finite-dimensional?
Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product.
Every finite-dimensional ...
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Larger possible chain of closed subspaces in the dual of a Banach space
In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces.
My question is the following. If $X$ is an ...
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Looking for Mackey's PhD thesis, "The subspaces of the conjugate of an abstract linear space"
I'm looking for a copy of George Mackey's PhD thesis, The subspaces of the conjugate of an abstract linear space (Harvard Univ., 1942), but am currently struggling to find one online, with the only ...
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Is the space $C_0^{k}(\Omega)$ a Montel space?
I asked this question in the MathStackExchange, but I think I'm not get any answer.
I'm trying to find a reference for the following result:
Theorem: Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ ...
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Analogue of $\ell^2(X)$ over an arbitrary Banach ring
Let $X$ be a set. Over the Banach fields $F=\mathbb{R}$ or $F=\mathbb{C}$ we can define the Banach space$$\ell^2(X)=\{\xi\colon X\to F\mid \sum_{x\in X}|\xi(x)|^2<\infty\}$$which satisfies a list ...
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Factorization systems for vector bundles
Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
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Does the uniform boundedness principle holds for multilinear maps as well?
This question has been motivated by weak* completeness of distributions.
According to the answer in the above post, any barrelled locally convex topological vector space $E$ satisfies the uniform ...
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If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?
In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space.
I wonder if the same result holds valid in infinite dimensions. More ...
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Topology of ${\mathcal D}(\Omega)$ (space of test functions)
I have seen two approaches to the topology of ${\mathcal D}(\Omega)$:
(i) Let $K$ be a compact subset of $\Omega$ and consider the subset ${\mathcal D}_K(\Omega)$ of test functions with support ...
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Reference request: an introduction to nuclear spaces
I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the ...
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Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
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Is the test function topology a Mackey topology?
I am a physicist, and I have lately been thinking about distributions as they appear in quantum field theory. In the standard development of the theory of distributions, one considers the space $C^{\...
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1
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When do the weak-star and compact convergence (compact-open) topology coincide on the dual of a Banach space?
In Measures Which Agree on Balls by Hoffmann-Jørgenson, it is claimed on page 323 that for an arbitrary Banach space $E$, if $\pi$ is the topology on $E^*$ of uniform convergence on compact subsets of ...
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The Fourier projection mappings $\{ P_N \}$ form an equicontinuous family of linear maps on $E'(S^1)$ as well?
Let $S^1=\mathbb{R}/\mathbb{Z}$ and define the Fourier projection operator $P_N$ for each $N \in \mathbb{N}$ as
\begin{equation}
P_N(f)=\sum_{n=-N}^N \langle f, e_n \rangle_{L^2} e_n
\end{equation}
...
- 3,745
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Why is a certain projective limit of weighted symmetric Fock space, namely $\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p)$, separable
I have a question regarding separability of a certain locally convex space.
Let $H_{\tau}:=H^{\tau_1}(\mathbb{R}^n,\tau_2(x)dx)$ the weighted Sobolev Hilbert space with $\tau_1 \in \mathbb{N}, \tau_2(...
