I've just read that for an almost-complex manifold $(M^{2n},J)$ the almost-complex structure $J:TM\to TM$ can be understood as a tensor field of type $(1,1)$.
I don't get it.
According to the section "Using tensor poducts" from this wikipedia article, a tensor $T$ of type $(1,1)$ would belong to $V\otimes V^*$, so in our case, $J_p\in T_pM\otimes T_pM^*$.
But $J_p$ is by definition a linear map from $T_pM$ to itself, so $J_p\in \text{End}(T_pM)$.
What am I missing?
If with respect to a coordinate system $(x^j)$ on $M$ you write $J(\partial_j) = \sum J^i_{~j}\partial_i$, then $J$ can be regarded as $$J = \sum J^i_{~j}\,{\rm d}x^j \otimes \partial_i,$$acting on vector fields by $$X \mapsto J(X) = \sum J^i_{~j}{\rm d}x^j(X)\partial_i.$$There is nothing particular about manifolds or almost-complex structures here. In general, in the level of vector spaces we have the (natural) isomorphism $$V^*\otimes W \ni f\otimes w \mapsto (v \mapsto f(v)w) \in {\rm Hom}(V,W),$$whose inverse can be described in terms of a basis $(e_i)$ of $V$ (with dual basis $(e^i)$) by $${\rm Hom}(V,W) \ni T \mapsto \sum e^i\otimes T(e_i) \in V^*\otimes W.$$