2
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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? $\endgroup$ – Arthur Sep 6 '18 at 6:37
  • $\begingroup$ @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. $\endgroup$ – D12ac Sep 6 '18 at 6:40
  • $\begingroup$ 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? $\endgroup$ – Arthur Sep 6 '18 at 6:50
  • $\begingroup$ 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 $\endgroup$ – Calvin Khor Sep 6 '18 at 6:55
  • $\begingroup$ @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. $\endgroup$ – D12ac Sep 6 '18 at 7:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.