> [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]]. Let $\Lambda^k$ be the [[Differentiable Functor|differentiable functor]] sending each [[Banach Space|Banach space]] to the space of bounded [[Alternating Tensor|alternating]] $k$-tensors, then $\Lambda^k(TX)$ is a [[Vector Bundle|vector bundle]] of class $C^{p - 1}$, and its [[Sections of Vector Bundles|sections]] are known as the **differential**/$k$-forms. The space of these sections $\mathbf{S}(\Lambda^k(TX))$ is denoted as $\Omega^k(X)$.