There are many elegant results on the dimensions of the simple representations of a finite group $latex {G}&fg=000000$, of which I would like to discuss a few today.

The final, ultimate goal is:

Theorem 1Let $latex {G}&fg=000000$ be a finite group and $latex {A}&fg=000000$ an abelian normal subgroup. Then each simple representation of $latex {G}&fg=000000$ has dimension dividing $latex {|G|/|A|}&fg=000000$.

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In this post we will use the Krein–Milman theorem together with the Hahn–Banach theorem to give another proof of the Stone–Weierstrass theorem. The proof we present does not make use of the classical Weierstrass approximation theorem, so we will have here an alternative proof of the classical theorem as well.

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# Pick Interpolation and Hilbert Function Spaces

The  original Pick problem is to determine, given $N$ points \lambda_{1},….,\lambda_{n}.

# Hahn-Banach Theorem

Hahn-Banach Theorem(Geometric Form)

Set K=R or C

Let V be a topological vector space over K.If A,B are non-empty convex disjoint subsets of V,then

1)If A is open, then there exists a continuous linear map p:V->K and t in R such that p(a)<t<p(b) for a in A and b in B

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