To introduce the theory of subspaces in the Hamilton spaces of higher order, $H$, it was necessary to solve several difficulties, because the classical theory of subspaces could not be applied. In almost all theories the $m$-dimensional subspace in the $n$-dimensional space was given by the introduction of $m$-parameters and $n-m$ normal vectors $N$, but the transformation of their coordinates was always a problem. Here, we introduce in $H$ two complementary family of subspaces $H_1$ and $H_2$. In this way we obtain the complicated coordinate transformations expressed in elegant matrix form in $H$, $H_1$ and $H_2$, and determine their connections. This method allows us to obtain the transformations of the natural bases $\bar B$, $\bar B_1$ and $\bar B_2$ of $T(H)$, $T(H_1)$ and $T(H_2)$ further $\bar B^*$, $B^*_1$ and $B^*_2$ of $T^*(H)$, $T^*(H_1)$ and $T^*(H_2)$. As the elements of the natural bases are not transforming as tensors the adapted bases $B$, $B_1$, $B_2$ of $T(H)$, $T(H_1)$, and $T(H_2)$ are introduced using the matrices $N$, $N_1$ and $N_2$, respectively. For the dual spaces $T^*(H)$, $T^*(H_1)$ and $T^*(H_2)$ the adapted bases are $B^*$, $B^*_1$ and $B^*_2$ formed with the matrices $M$, $M_1$ and $M_2$, respectively. It is proved that $N$ and $M$, $N_1$ and $M_1$, $N_2$ and $M_2$ are inverse matrices to each other if $B^*$ is dual to $B$, $B^*_1$ is dual to $B_1$ and $B^*_2$ is dual to $B_2$. The main result is the construction of adapted basis $B'=B_1\cup B_2$ and $B^*{}'=B^*_1\cup B^*_2$ of $T(H)$ and $T^*(H)$ in such a way that the elements of $B'$ and $B^*{}'$ are transforming as tensors and the tensor from space $H$ can be decomposed as a sum of projections on $H_1$ and $H_2$. It is obtained by the determination of the relations between $N$, $N_1$ and $N_2$ further between $M$, $M_1$, and $M_2$. This very important result allows us to study the connections, torsion and curvature tensors, Jacobi fields, sprays and other invariants in the subspaces and surrounding space and determine their relations which will be done later on.