Basic calculus on topological fields

Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $ \mathbb Q_p$). 1) Let $f: K^n \to K$ be a function such that its differential $Df$ is constant $0$. Is $f$ constant on … Read more

Locally compact vector space over a finite field

In the wikipedia article titled “topological vector space”, there is a line saying the following. “Let K be a locally compact topological field, for example to real or complex numbers. A topological vector space over K is locally compact if and only if it is finite-dimensional, that is, isomorphic to Kn for some natural number … Read more

A characterization of nuclear functionals in terms of continuity with respect to some special topologies on B(X)B(X)?

I think, nuclear functionals on the space of operators B(X) (on a Banach space X) must have a characterization in terms of some special continuity. I would be grateful if somebody could help me with the following hypothesis. First, a nuclear functional on B(X) (where X is a Banach space) is a linear functional u:B(X)→C … Read more

Pointwise vs. local homotopy equivalences of continuous and smooth complexes of real vector bundles

Let $(E^\bullet,d_E)$ and $(F^\bullet,d_F)$ be two complexes of real vector bundles on a topological manifold $X$, and let $f^\bullet\colon E^\bullet\to F^\bullet$ be a morphism of complexes, i.e. a collection of degreewise morphism of vector bundles $f^n\colon E^n\to F^n$ compatible with the differentials. Then at every point $x$ of $X$ we have a morphism $f^\bullet_x\colon E_x^\bullet\to … Read more

Topological localisations of algebras and solidification

Let A be a (topologically discrete) commutative ring and consider the topological ring A((z)). Let A((z))_ denote the corresponding condensed ring, and let a(z) be a non-zero element of A((z)). We can form the localisation A((z))_[a(z)−1] inside condensed rings as (−)⊗Z[t]Z[t,t−1]. Question 1. Is this isomorphic to the condensed ring corresponding to the localisation, A((z))[a(z)−1], … Read more

Finite generation of vector identities

This question is partially motivated by, although that question may not be appropriate for MO. Consider the set E of all valid equalities (without parameters) over the two-sorted structure (R,R3) in the language consisting of symbols for: the arithmetic operations addition, subtraction, and multiplication on R; the vector operations of vector addition, vector subtraction, … Read more

Can you pair Hs(Ω)H^s(\Omega) and H−s(Ω)H^{-s}(\Omega) on a domain Ω\Omega?

Consider the fractional Sobolev spaces on Rn Hs(Rn):={u∈S′(Rn):(1+|ξ|2)s/2ˆu∈L2(Rn)}. Let Ω be any open subset of Rn, we then define Hs(Ω):={u∈D′(Ω):u=RΩU,U∈Hs(Rn)}, where RΩU denotes the restriction of distributions. Note that Hs(Ω) is defined equivalently by factoring those elements of Hs(Rn) which have no support on Ω. Let Hs(Rn) and Hs(Ω) be equipped with the canonical scalar … Read more

Productivity of certain sequential subcategories of topological vector spaces

Consider the usual sequential modifications of topologies (spaces) in the categories of topological spaces Top, topological vector spaces TVS and locally convex spaces LCS : Let X be a real vector space. All of the occurring topologies are assumed to be Hausdorff. If τ is a topology, linear topology, locally convex topology on X then … Read more

Approximation of the identity by finite range functions in topological vector spaces

Let $X$ be a topological vector space. Assume that there exists a sequence $\phi_n:X\to X$ of finite range measurable functions with $\lim\phi_n(x)=x$ for every $x\in X$. Can we concluded there exists a sequence $\{X_n\}$ of subsets of $X$ with $X=\cup X_n$ such that $X_n$’s are all relatively second-countable? Note that the answer will be negative … Read more

Real eigenvectors of complex matrices

Let $A$ be a nonsingular complex $(3 \times 3)$-matrix (that is, an element of $\mathrm{GL}_3(\mathbb{C})$). Then what are some of the best-known criteria which guarantee $A$ to have real eigenvectors ? (I am also interested in the same question for nonsingular complex $(n \times n)$-matrices with $n \geq 2$, but my main target is the … Read more