Compactly supported distributions as a projective G-module

For a Lie group G and a locally convex space V let E(G,V) be the locally convex space of smooth functions from G to V, and accordingly E′c(G,V) the space of compactly supported distributions. A G-module V is called differentiable if v→(g→gv) defines a continuous map from V to E(G,V) for all v∈V. In particular … Read more

point-wise approximation of the identity in hereditary Lindelof spaces

Let X be a topological vector space. Assume that there exists a sequence of finite range measurable functions ϕn:X→X with lim. Q. Can we concluded that X is hereditery Lindelof? By an interesting example Taras Banakh proved that: X is not necessarily written by countable union of second countable subsets (see here). Answer AttributionSource : … 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

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

Hahn-Banach restricted to a pre-dual

If V is a locally convex topological space, the Hahn-Banach theorem shows that a continuous linear functional on a closed subspace can be extended to a continuous linear functional on all of V, that is, an element of the dual space V∗. If it is known that V=U∗, it has a locally convex pre-dual U⊆V∗, … Read more

Existence of a countable linear combination with positive coefficients

Consider a (Hausdorff and complete) locally convex topological vector space V and a countable subset (vk)∞k=1⊂V of non-zero vectors. (∗) Under what conditions on this subset are we guaranteed the existence of a sequence of positive real numbers (αk)∞k=1, such that the series ∑∞k=1αkvk converges (that is, the sequence of its partial sums) converges to … Read more

Reference for Choquet-like theorem

While reading a paper, I encountered the following statement: Let K be a convex compact subset of a locally convex topological vector space. If μ∈P(K) is a Radon probability measure on K, then there is a unique point xμ∈K such that ∫Kfdμ=f(xμ) for every continuous, affine, real-valued function f:K→R. Moreover, the map P(K)→K:μ↦xμ is a … Read more

On Köthe sequence spaces

I asked this a week ago at math.stackexchange, but without success. As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his “Locally convex spaces” defines it as the space Λ(P) of sequences λ:N→C satisfying the condition ∀α∈P∞∑n=1αn⋅|λn|<∞, where P is an arbitrary … Read more

pointwise convergence to the identity

Let X be a separable topological vector space with size (cardinal number) no larger than c. Does there exist any sequence of finite rank linear maps ϕn:X→X pointwise converging to the identity mapping id:X→X? Answer Just to chat, an easier counterexample is X:=Lp(R) for 0≤p<1, a complete metric separable TVS. The identity map can’t be … Read more

L∞L^{\infty} as colimit

I recently read a result (in Jarchow’s book) that any ultrabornological space can be expressed as a colimit (in the category LCS) of Banach spaces. My question is the following. Let μ be a finite measure on R. Is there an uncountable set of pi∈[0,∞) and finite Borel measures μi on R for which one … Read more