# Definition of Dual Vector Space used to define a Tensor

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?

• This may have to do with the dimension of $V$ (if I have all my facts straight). Isn't any linear functional on a finite-dimensional (normed) vector space bounded? – Arthur Sep 6 '18 at 6:37
• @Arthur Yes, on a finite dimensional vector space $V$, every linear functional is bounded. On such a $V$, both definitions of the dual would be equivalent. – D12ac Sep 6 '18 at 6:40
• So... if we're in a setting where $V$ is assumed to be finite-dimensional (like, say, the tangent space over a point on a manifold) then we don't need to specify boundedness, while in general we do. Maybe that's what's going on with your two definitions? – Arthur Sep 6 '18 at 6:50
• I believe it depends on the application. In functional analysis the dual is almost always the continuous dual, but I imagine that the algebraic dual is used in other settings. See en.wikipedia.org/wiki/Dual_space#Continuous_dual_space – Calvin Khor Sep 6 '18 at 6:55
• @Arthur What irks me is (on infinite dimension) the need to defined a norm on a vector space to define a tensor. I don't even know if this statement is correct or not. – D12ac Sep 6 '18 at 7:05