User:Alsosaid1987/sandbox

$$(f^*\omega)(p)=(f_*)^*(\omega(f(p)))$$

For instance, take a space of real-valued functions $$f\in\mathcal{F}$$, $$f:V\to\mathbb{R}$$ and fix an element of the domain $$v_0\in V$$. Suppose we wanted a map $$\Phi_{v_0}:\mathcal{F}\to\mathbb{R}^{\mathcal{F}}$$ that takes $$f\in\mathcal{F}$$ to a functional defined by $$f\mapsto f(v_0)$$. We can use the bracket notation and write $$\Phi_{v_0} = [v_0,\cdot]$$, where the dot notation is used to mean that $$[v_0,\cdot]$$ is the map $$f\mapsto [v_0,f]$$.

$$d\circ d=0$$