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 ${\mathcal{A}}_{\varsigma }^p$ in this paper whose object is a triple $(N,v,\delta )$ given by an $(A,B)$-bimodule $N$, a special element $v\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $\delta :N^{{\oplus }_p{2}^{\dim A}}\to N$ and each morphism $(N,v,\delta )\to (N^{\mathrm{\prime }},v^{\mathrm{\prime }},{\delta }^{\mathrm{\prime }})$ is given by an $(A,B)$-homomorphism $\theta :N\to N^{\mathrm{\prime }}$ such that $\theta (v)=v^{\mathrm{\prime }}$ and ${\delta }^{\mathrm{\prime }}{\theta }^{\oplus 2^{\dim A}}=\theta \delta$ hold. We show that ${\mathcal{A}}_{\varsigma }^p$ 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 ${\mathcal{A}}_{\varsigma }^p$ starting with this initial object.