How can we define a covariant $k$-tensor this way if tensors are already defined by the tensor product?

Question

From Lee's Intro to Smooth Manifolds:

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If we take $k$ covectors, $\varepsilon^{i_1}, \dots, \varepsilon^{i_k},$ then the tensor product is defined by $$\varepsilon^{i_1}\otimes \cdots \otimes \varepsilon^{i_k}(v_1, \dots, v_k)=\varepsilon^{i_1}(v_1)\cdots \varepsilon^{i_k}(v_k).$$

So, how can we define a covariant $k$-tensor as in the image above?

Answer 1

You define it as shown, as there is no conflict of definitions. Perhaps the $k = 2$ case can render a bit more readability.

Although

$$ \varepsilon ^ {i_1} \otimes \varepsilon ^ {i_2}, \varepsilon^{(i_1,i_2)}:V^2\rightarrow \mathbb{R}$$

have the same domain and codomain, they are defined entirely differently, as shown in your post:

$$\varepsilon^{i_1}\otimes\varepsilon^{i_2}(v_1,v_2) = \varepsilon^{i_1}(v_1) \varepsilon^{i_2}(v_2)$$

$$\varepsilon^{(i_1,i_2)}(v_1,v_2) = \varepsilon^{i_1}(v_1)\varepsilon^{i_2}(v_2) - \varepsilon^{i_1}(v_2)\varepsilon^{i_2}(v_1)$$