Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the linear structure is defined as follows: $$(a\Phi+b\Psi)(u_1,\cdots,u_r):=a\cdot \Phi(u_1,\cdots,u_r)+b\cdot \Psi(u_1,\cdots,u_r),\quad\forall \Phi,\Psi\in\mathfrak T^r (\Bbb R^m)$$ It is easy to check that elements from $\mathfrak T^r (\Bbb R^m)$ is closed under linear combination, and the other axioms to construct a linear space are fulfilled as well, except that I'm having trouble finding its unique zero element!
I'm aware that any such tensor under a certain cobasis $(g_1,\cdots, g_m)$ can be written into the "simple form" $$\Phi=\Phi(g_{i_1},\cdots,g_{i_r})\cdot g^{i_1}\otimes\cdots\otimes g^{i_r}$$ If $\Phi$ maps any combination of $r$-$g_i$s onto $0$ then it can surely be identified as the zero element, but how to show such a zero tensor is unique? How to show there is not another way to define a zero tensor? The summation is too complicated here and I can't figure it out. Can u explain it to me ?
Thanks in advance!