Consider a connected topological space $$X$$ with point $$x$$ in and let $$K$$ be field the discrete topology. We study Tannakian category of finite-dimensional (flat) vector bundles on its dual $$\pi(X,x)$$ respect to fiber functor . The maximal pro-étale quotient is étale fundamental group studied by Kucharczyk Scholze. For well-behaved spaces, pro-algebraic completion ordinary group. obtain ...

we define a new function-valued inner product on l2(g), called ?-bracket product, where g is a locally compact abelian group and ? is a topological isomorphism on g. we investigate the notion of ?-orthogonality, bessel's inequality and ?-orthonormal bases with respect to this inner product on l2(g).