I am studying the math of tensors, I have an understanding of the concepts of covariance, contravariance, dual spaces, Einstein notation and so on. I am a bit confused about the notation though. My quick questions:

• Is $v^ie_{i}$ a tensor? ($v$ are the components of a vector and $e$ are the basis vectors)
• Is $v^iv_{i}$ a tensor?
• If both the above are tensors, how do you distinguish between the two? If not, isn't this an abuse of notation?
• When talking about a tensor, does one usually mean $v^iv_{i}$, $v^ie_{i}$, or both? (I guess this depends on the answer to the first two points)