Normed representations of weight quivers
Let $A$ and $B$ be two tensor rings given by weight quivers. We introduce norms for tensor rings and $(A,B)$-bimodules, and define an important category $\mathscr{A}^p_ς$ in this paper whose object is a triple $(N,v,δ)$ given by an $(A,B)$-bimodule $N$, a special element $v\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $δ: N^{\oplus_p 2^{\dim A}} \to N$ and each morphism $(N,v,δ) \to (N',v',δ')$ is given by an $(A,B)$-homomorphism $θ: N\to N'$ such that $θ(v)=v'$ and $δ' θ^{\oplus 2^{\dim A}} = θδ$ hold. We show that $\mathscr{A}^p_ς$ has an initial object such that Daniell integration, Bochner integration, Lebesgue integration, Stone--Weierstrass Approximation Theorem, power series expansion, and Fourier series expansion are morphisms in $\mathscr{A}^p_ς$ starting with this initial object.