I am a little confused about a definition of tensors. If I define a space of all tensors as: $$T_l^k(V) \equiv \underbrace{V\otimes\ldots\otimes V}_l\otimes \underbrace{V^*\otimes \ldots \otimes V^*}_k$$, a tensor is then a member of this space and they are defined as a multilinear maps from $\underbrace{V^* \times \ldots V^*}_l \times \underbrace{V \times \ldots\times V}_k$ (or I think more precisly, it should be written $V^* \times \ldots V^* \times {V^*}^* \times \ldots\times {V^*}^*$) to $\mathbb{R}$.
Thus for each set of $(v_1^*,\ldots,v_l^*,v_1,\ldots,v_k)\in V^* \times \ldots V^* \times V \times \ldots\times V$, $T(v_1^*,\ldots,v_l^*,v_1,\ldots,v_k)$ is a number.
But then my notes give examples of various obvious tensors like vectors, linear forms, bilinear forms. But then it says (also wikipeadia and just any other source i can find) that the tensor of type $(1,1)$ is a linear map from $V$ into self: $f:V\to V$ but something is wrong here because the tensor should produce a number not map a vector space onto itself. Or is this the same situation as with double dual space that people write $V$ instead of ${V^*}^*$ even though that is like comparing apples and pork, they are just completely different thing (even though there is an isomorphism between them)