A variant of this question has been asked before, but not the specific one I had in mind.
A tensor is defined as a multi-linear map from $l$ copies of a dual vector space $V^*$ and $m$ copies of a vector space $V$ to the field $K$ of the vector space. How has the dual vector space been defined? I have seen two definitions for the dual vector space- the set of all linear functionals and the set of bounded linear functionals (which would require a norm to be defined to define a tensor(!?)).
If we were to define a norm on $V$, would the definition of the dual space, and hence a tensor, change?
Link to a similar question: Dual space - bounded functionals?