# 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.

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### $M\otimes _{\mathbb{Z}_p}\mathbb{Q}_p$ is Banach if $M$ is $p$-adically complete

I was reading the paper On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms of Emerton, in which at page 15 the following fact is claimed (in my notation).
Let $p$ ...

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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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### 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}
...

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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(...

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### Duality and compactness for pro vector spaces

I have a somewhat basic question which I haven't been able to piece together from the literature.
Background. We work over a field $\bf{k}$. Consider the category, $\bf{Pro}_{k}$, of pro- vector ...

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### Do radially bounded sets form a bornology?

We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, ...

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### Minimal F-semi-norms

There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\...

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### Pointwise convergence and disjoint sequences in $C(K)$

Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with ...

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### Goldstine theorem in quasi-Banach spaces

A classical theorem of Goldstine is the following: Let $X$ be a Banach space and $J \colon X \to X''$ the natural inclusion. Then $J(B_X)$ is $\sigma(X'', X')$-dense in $B_{X''}$, where $B_Y$ is the ...

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### Equality of topologies in the spaces of section of a vector bundle

In this notes Geometric Wave Equations by Stefan Waldmann at page 7 he has
Let $E \longrightarrow M$ be a vector bundle of rank $N$. For a chart $(U, \psi)$ we consider a compact subset $K \subseteq ...

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### Quasi-completion of a locally convex space as a space of linear functionals on its dual

A locally convex topological vector space is said to be quasi-complete if its closed bounded subsets are complete. Given a locally convex space $E$, one can show the existence of a Hausdorff quasi-...

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### Decomposition of weak* convergent nets into positive weak* convergent nets

Let $F$ be an order unit Banach space with order unit $e$ and topological dual space $F^*$ ordered by the dual cone. Let $E\subset F^*$ be a closed subspace that separates points of $F$ and such that ...

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### Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points

Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$
\begin{align}
\max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...

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### Parametrization of topological algebraic objects

There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...

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### Schwartz kernel theorem for restricted operators

Let $(M,g)$ be a smooth Riemannian manifold. The celabrated kernel theorem of Schwartz shows that for any linear and continuous operator $A:C_{c}^{\infty}(M)\to C^{\infty}(M)$, there exists a ...

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### Is $B \cap L^2\big( (0,T) \times (0,1)\big)$ closed in $L^2\big( (0,T) \times (0,1)\big)$?

I need help proving that the set $B \cap L^2\big( (0,T) \times (0,1)\big)$ is a closed subset of $L^2\big( (0,T) \times (0,1)\big)$, where $B$ is defined as:
$$B=\Big\{x \in L^{\infty}\big(0,T;L^1(0,1)...

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### Neighborhoods of idempotents in topological inverse semigroups

In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse ...

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### Infinite-dimensional BRST reduction

Fix a base field $k$. First let me loosely describe the BRST reduction in the finite-dimensional setting. For a finite-dimensional Lie algebra $\mathfrak{n}$, we can form the Clifford algebra $\...

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### Complex interpolation of subspaces

Let $(X_0,X_1)$ be an interpolation couple of Banach spaces. Using complex interpolation we can form Banach spaces $X_\theta:=(X_0,X_1)_\theta$ where $0<\theta<1.$ Let $E_\theta\subseteq X_\...

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### Topological vector spaces in direct sum

A year ago, I asked this question here at Mathematics Stackexchange, but no one there managed to answer it. So I am elevating it to MathOverflow.
This question had emerged as an offshoot of a bigger ...

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### Reference for Schwartz kernel theorem on vector bundles

In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has
Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...

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### Projective limit of Fréchet reflexive spaces

I am reading this paper, constructing spaces of functions and distributions with exponential growth on Fréchet nuclear spaces and their dual.
Un théorème de dualité entre espaces de fonctions ...

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### Fréchet and DF spaces

Is there a canonical way to make a DF-space Fréchet while keeping the same vectorial structure? Or the converse? I've been looking in the classical books for locally convex spaces but haven't found ...

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### Is identity map on the space of smooth maps smooth?

I'm curious about the identity map on the space of all smooth maps (between two locally convex topological vector spaces in the sense of convenient calculi) when we equip the space with different ...

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### Is the strong topology the strongest?

Let $X$ be a topological vector space. We know that the weak topology $\sigma(X,X^*)$ is the weakest locally convex topology in $X$ that make every $f \in X^*$ continuous. Consequently, if $\tau$ is ...

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### Complemented subspaces in a dual Banach space

Let $Y$ be a complemented subspace in a dual Banach space $X$. Is it true that $Y$ is itself isomorphic to a dual?
This is the case of a $w^*$-closed subspace $Y$, but a complemented subspace of $X^*...

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### Under what conditions does a continuous linear map map a closed subspace to a closed subspace?

Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$?
It is obviously satisfied if $W$ is ...

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### On the equality $\{f\in C(X), f|_A=0\}+\{f\in C(X), f|_B=0\}=\{f\in C(X), f|_{A\cap B}=0\}$

Let $X$ be a Tychonoff space, let $A,B\subset X$ be closed. Let $J_A$ be the set of all continuous on $X$ real-valued functions which vanish on $A$.
For which $X$'s is it true that $J_A+J_B=J_{A\cap ...

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### Morphism in commutative square strict?

Let $G,H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism.
Then $f$ is said to be strict if $G/\mathrm{Ker}(f) \cong \mathrm{Im}(f)$ is an isomorphism of topological ...

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### Weakly sequentially closed convex cone which is not weakly closed

Let $V$ be the real vector space of finitely supported functions $f: \Omega\to \mathbf{R}$ such that $\sum_\omega f(\omega)=0$, where $\Omega$ is a given uncountable set.
Endow $V$ with the weak ...

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### Consequences of having unbounded points in a bornology

For a set $X$ a bornology $\mathcal{B}$ is essentially an ideal in the power set $\mathcal{P}(X)$. Many sources including Wikipedia state additional property that $X = \bigcup \mathcal{B}$. Call it a ...

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### Splitting of a topological vector space (TVS) into an (a) countable sum and (b) direct integral of subspaces

I thought that this would be a simpe question, and placed it here at the Mathematics Stackexchange. Now have to elevate it to Mathoverflow.
LANGUAGE
TVS = topological vector space. Any subspace of a ...

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### Are there "pathological convex sets" over ultravalued fields of char 2?

In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...

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### $\sigma$-compactness of some locally compact Hausdorff topological groups

Is the topological group $(\mathbf{Q}_p/\mathbf{Z}_p)^{\oplus k}$, $k\ge 1$, a $\sigma$-compact topological group when endowed with its natural $p$-adic topology?
More generally, I'm looking for a ...

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### Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?

I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below).
It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be
written as an ...

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### Generalized Jensen's inequality for positively homogeneous functions

The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...

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### An explicit description for a certain type of infinite-dimensional homogeneous polynomials

This is a side question from Infinite-dimensional "algebraic varieties".
Denote by $X_p$ ($1 \le p \le \infty$) the Banach spaces of complex sequences with finite $p$-norm and limit $0$. ...

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### Fundamental result on the projective tensor product of sections of a vector bundle

Definition Let $E_{1} \xrightarrow{\pi_{1}} M_{1}$, $E_{2} \xrightarrow{\pi_{2}} M_{2}$ be two vector bundles over $M_{1}$ and $M_{2}$ with fibers $V_{1}$, $V_{2}$ respectively. The exterior tensor ...

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### Looking for an electronic copy of Trèves and Martineau books

I would like to know if anyone has an electronic copy of the following books:
MR0177291 Martineau, A.; Trèves, F. Éléments de la théorie des espaces vectoriels topologiques et des distributions. Fasc....