>>9284631

>single book

No.

>Banach algebras

Any functional analysis.

>Affine Hecke algebra

Any mathematical text on conformal field theory.

>Exterior algebra

Any differential topology book.

>Koszul algebra

Any homological algebra text.

>Poisson algebra

Any classical mechanics text.

>Weyl algebra

Any mathematical text on quantum field theory.

>>9284936

Let [math]X[/math] be a [math]G[/math]-space and denote the action of [math]G[/math] on [math]X[/math] by [math]g \mapsto (x \mapsto g(x))[/math], then the [math]G[/math]-quotient is the space of orbits [math]\mathcal{O}_x = \{y \in X\mid y = g(x) \}[/math].

>>9287959

Let [math]f\in\mathcal{S}_c[/math] be a test function and suppose [math]\phi[/math] satisfies the equation [math]\phi'' + \phi = 0[/math], then [eqn]\int_K dxf(x)\phi(x) = -\int_K dxf(x)\phi''(x) \rightarrow_{x \rightarrow nx} -\frac{1}{n} \int_K d(nx) f(nx)\phi''(nx) = -\frac{1}{n}\int_K dx f(x)\phi''(x) = -\frac{1}{n}\int_K dx f(x)\phi(x) \rightarrow 0[/eqn]

Remember that scalings of compact sets are still compact so [math]f_n(x)\equiv f(nx) \in \mathcal{S}_c[/math].