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If a topological vector space $X$ is not locally convex, then it usually has not non-zero linear continuous functionals, and this means that there are no non-trivial continuous characters on $X$. For …

Let $G$ be a locally compact group, $K$ a compact subgroup in $G$, and $\mu_K$ the normalized Haar measure on $K$: $$ \mu_K(K)=1. $$ Let us denote by $\widetilde{\mu_K}$ the measure on $G$ defined as …

I am sorry, I have realized that the answer is "yes", and this is simple. The proof is the following. Suppose this is not true. Then we can find a locally compact group $G$ which is not locally Euclid …

I asked this 11 days ago at MSE, but there was no answer, I hope people here could help. Let $G$ be a locally compact group, and $X$ a Hilbert space. A unitary representation $\varphi:G\to B(X)$ is sa …

I would say that this is not well-known: It is well-known that a Banach space $V$ is always Pontryagin-reflexive, i.e. the natural map $V\to \text{Hom}_\mathbb{R}(\text{Hom}_\mathbb{R}(V, \mathbb{R}) …

I strongly recommend you to read the François Bruhat paper, that Osborne cites. For an arbitrary locally compact (not necessarily abelian) group $G$ Bruhat defines smooth function $\varphi:G\to{\mathb …