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 Posted: Tue Apr 03, 2007 8:39 pm
I just started taking a course on Multivariable Analysis and after three days of "review" I am very confused. I think I understand what a dual of a vector space is, but what is dual basis expansion and what does it do? What is the double dual of a vector space?
Is this really stuff an upperclassman undergraduate like me should know already? @_@;; I mean, the grad students in the class seem to know it already, so maybe I'm missing something.
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Posted: Wed Apr 04, 2007 10:21 am
Are you talking about the algebraic dual space or the continuous dual? Basically, is it a topological vector space or just a vector space? Actually, if your vector space is finite-dimensional then it doesn't matter, as all linear functions are continuous; only in infinite dimensional spaces can you have discontinuous linear functions.
The dual basis is just a basis of the dual space, often matched up with a basis of the vector space so that if your vector space basis is v_i and your dual basis is v_i*, you get that v_i*(v_j) = 1 iff i = j and 0 otherwise. Basically it's just a way to calculate what you get when you apply dual elements to vectors.
The double dual is the dual of the dual space, i.e. the set of linear functions that take linear functionals on V as arguments and return elements of the base field, i.e. if w* is an element of the dual space and v** an element of the double dual, then v**(w*) is an element of the underlying field F.
If the vector space is finite-dimensional, there is a canonical isomorphism P between the double dual and the original vector space given by (P(v))(w*) = w*(v).
Have you done much linear algebra? This is what you'd get in a course on abstract linear algebra.
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Posted: Wed Apr 04, 2007 11:26 am
Layra-chan Are you talking about the algebraic dual space or the continuous dual? Basically, is it a topological vector space or just a vector space? Actually, if your vector space is finite-dimensional then it doesn't matter, as all linear functions are continuous; only in infinite dimensional spaces can you have discontinuous linear functions. The dual basis is just a basis of the dual space, often matched up with a basis of the vector space so that if your vector space basis is v_i and your dual basis is v_i*, you get that v_i*(v_j) = 1 iff i = j and 0 otherwise. Basically it's just a way to calculate what you get when you apply dual elements to vectors. The double dual is the dual of the dual space, i.e. the set of linear functions that take linear functionals on V as arguments and return elements of the base field, i.e. if w* is an element of the dual space and v** an element of the double dual, then v**(w*) is an element of the underlying field F. If the vector space is finite-dimensional, there is a canonical isomorphism P between the double dual and the original vector space given by (P(v))(w*) = w*(v). Have you done much linear algebra? This is what you'd get in a course on abstract linear algebra. Or the basics of differential geometry sweatdrop
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Posted: Wed Apr 04, 2007 1:42 pm
Layra-chan Have you done much linear algebra? This is what you'd get in a course on abstract linear algebra. Yeah, I spoke with my prof and that's basically what he said too. I didn't take the more abstract version of linear algebra. *headdesk* Know any good textbooks or online sources where I can quickly play catch-up?
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