• $v^ie_i$ is a scalar.
• $v_iv^i$ is a scalar.
• They're both scalars
• When talking about tensors, we usually just mean the components $v_i$ (In this case, the $v_i$s transform as a vector)

We usually have a vector space with basis $e_i$ and each element of the vector space $A$ can be expanded as $\vec{V} = v^ie_i$. The $v^i$ completely determines $\vec{V}$ in a given basis, and it is these components that transform as a vector. Note that under a change of basis $\vec{V}$ itself remain unchanged and that's how we get the rules of transformation for $v^i$ from that of $e_i$.

We get dual space by realizing that the space of all linear functionals $T:A\to \mathbb{R}$ themselves form a linear vector space. As a result we can expand each element of $T$ along a basis $e^j$ defined by $e^je_i = \delta_{ij}$. This is the dual space.

We get higher order tensors by taking the tensor product $A^n \otimes T^m$ and each element $r$ can be expanded along the basis of this space. The coefficients are what transforms as a tensor and that's what we're interested in.

Sean Carroll's GR book has a nice section on the whole construction of the dual space and tensor product spaces in a crisp manner.

• $v^ie_i$ is an element of the vector space, but it's not a vector in the same sense as $v_i$. This is because we define vectors/tensors based on their transformation properties.
• $v_iv^i$ is a scalar.
• Points addressed above
• When talking about tensors, we usually just mean the components $v_i$ (In this case, the $v_i$s transform as a vector)

We usually have a vector space with basis $e_i$ and each element of the vector space $A$ can be expanded as $\vec{V} = v^ie_i$. The $v^i$ completely determines $\vec{V}$ in a given basis, and it is these components that transform as a vector. Note that under a change of basis $\vec{V}$ itself remain unchanged and that's how we get the rules of transformation for $v^i$ from that of $e_i$.

We get dual space by realizing that the space of all linear functionals $T:A\to \mathbb{R}$ themselves form a linear vector space. As a result we can expand each element of $T$ along a basis $e^j$ defined by $e^je_i = \delta_{ij}$. This is the dual space.

We get higher order tensors by taking the tensor product $A^n \otimes T^m$ and each element $r$ can be expanded along the basis of this space. The coefficients are what transforms as a tensor and that's what we're interested in.

Sean Carroll's GR book has a nice section on the whole construction of the dual space and tensor product spaces in a crisp manner.