
Covectors Are Different Than Vectors, Usually - pizza
http://infohost.nmt.edu/~parendt/Phys509/notes/2diracnot/node4.html
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gus_massa
> _(Fortunately, when a space of covectors is considered to be a vector space
> of its own, their covectors are members of the original vector space, so we
> never have too many vector spaces to worry about at one time.)_

Bad news. This is only valid in many interesting cases, but not in all cases.
For example if you take

* "c_0" = the space of all sequences which converge to 0, with the supreme norm

* "l^1" = the space of all the sequences with and absolute sum

* "l^∞" = the space of all the bounded sequences

then the dual of c_0 is l^1 and the dual of l^1 is l^∞. An important general
result is that the original space is included in the double dual, in this case
c_0 is included in l^∞.

