# Tagged Questions

Locally compact groups are among the main objects of study of abstract harmonic analysis. They arise e.g. in number theory, abstract Fourier analysis, representation theory, Lie theory, operator theory, etc. Their main property is the existence and uniqueness of Haar measure. Please combine with ...

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### closed subset of locally compact

A space $X$ is said locally compact if for any $x\in X$ and for any neighbourhood $U$ of $x$ there is a compact neighbourhood $V$ such that $V\subseteq U$. Does closed subset of locally compact is ...
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### sequential continuity and countinuity

When we have two topological spaces, $\left(X, \tau_X\right)$ and $(Y, \tau_Y)$ it is easy to check that for $f: X \rightarrow Y$ continuity implies sequential continuity. I'm wondering what do we ...
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### Intersection between a compact and a locally compact set

I'm trying to understand Rudin's proof of Pontryagin duality theorem, but I still haven't undersood an argument. (Fourier analysis on groups, p29) Let $G$ be a group and denote $\Gamma =\widehat{G}$ ...
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### Stabilizer, Cosets, homeomorphism and Compact groups : proving things in The Structure of Compact Groups by Hofmann and Morris

I'm currently struggling trying to prove a few things in the book The Structure of Compact Groups by Hofmann and Morris. The first one would be Proposition 1.10.i (or E1.4) : If the topological ...
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### Transpose in ${SL}(2,\mathbb{R})$

Let $SL(2, \Bbb{R})$ denote the group of special invertible $2\times 2$ matrices over $\mathbb{R}$. As a locally compact group, the Haar measure of $SL(2, \Bbb{R})$ is computable through Iwasawa ...
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### Countable LCA groups

Is it true that a countable LCA group can only be discrete ? This question is related to a comment here : A theorem on LCA group - is the uncountability necessary?
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### On the structure of non discrete locally compact topological (non-necessarily commutative) complete fields.

There is totally classic result about the structure of non discrete locally compact topological (non-necessarily commutative) fields $K$, whose proof uses the existence of the Haar measure on the ...
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### Reduced $C^*$-algebra of a direct product of locally compact groups

Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as ...
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### Induced measure on dual group and kernel of fourier transform

Let $G$ be a locally compact Hausdorff abelian group (LCA). Then the Fourier transform gives a map from functions on G to functions on $\hat G$ f\mapsto \hat f(\xi) = \int_G f(x)\Psi(x,\xi) \mu_G(x)...
### Integral of $|f|$ outside a compact set
Let $G$ be a locally compact group. Given $f\in L^1(G)$ and $\epsilon>0$, how to show that there is a compact set $K\subset G$ such that $\int_{G\setminus K}|f|<\epsilon$?