Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $u\colon L\to M$ be a linear map of locally convex linear topological vector spaces. Assume that $u$ is sequentually continuous, i.e. maps convergent sequences to convergent ones. (This notion is formally weaker that the usual topological continuity in the case of non-metrizable spaces.) Let $L_0\subset L$ be a topologically dense linear subspace. Assume that $u|_{L_0}\equiv 0$.

QUESTION: Does it follow that $u\equiv 0$?

I am interested in rather concrete examples of spaces: spaces of generalized functions on smooth manifolds (say $R ^n$) with the wave-front set contained in a given closed set.

share|improve this question

1 Answer 1

up vote 8 down vote accepted

Take $c(\Gamma)$ with $\Gamma$ uncountable under the topology of pointwise convergence. $c_0(\Gamma)$ is dense but not sequentially dense. Let $u$ be the linear functional that vanishes on $c_0(\Gamma)$ and is one at $1_\Gamma$.

share|improve this answer
3  
Product space $\ \mathbb R^A\ $ for uncountable $\ A\ $ works like this too (but my poor brainy malfunctions from time to time, too often). –  Włodzimierz Holsztyński Oct 1 '13 at 18:50
    
Bill, good example. Thanks. –  semyon alesker Oct 2 '13 at 3:36

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.