Summary: Let \(\mathcal V\) be a Banach space over a field \(\mathcal F\). A \(\overrightarrow{G}\)-flow is a graph \(\overrightarrow{G}\) embedded in a topological space \(\mathcal S\) associated with an injective mappings \(L:u^v\to L(u^v)\in\mathcal V\) such that \(L(u^v)=-L(v^u)\) for all \((u, v)\in X (\overrightarrow{G})\) holding with conservation laws \[ \sum_{u\in N_{G}(v)}L(v^u) = 0\ \text{for all } v\in V (\overset{\rightarrow}{G}), \] where \(u^v\) denotes the semi-arc of \((u, v)\in X(\overrightarrow{G})\), which is a mathematical object for things embedded in a topological space. The main purpose of this paper is to extend Banach spaces on topological graphs with operator actions and show all of these extensions are also Banach space with a bijection with a bijection between linear continuous functionals and elements, which enables to solve linear functional equations in such extended space, particularly, solve algebraic, differential or integral equations on a topological graph, find multi-space solutions on equations, for instance, the Einstein’s gravitational equations. A few well-known results in classical mathematics are generalized such as those of the fundamental theorem in algebra, Hilbert and Schmidt’s result on integral equations, and the stability of such \(\overrightarrow{G}\)-flow solutions with applications to ecologically industrial systems are also discussed in this paper.