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nomadreid
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- In an non-orthonormal vector space V with inner product and its dual space V*, I presume the two situations are the same: (i) to an f in V* corresponds v in V, by Riesz, (ii) to a v in V corresponds to its dual f in V. There are some details here that confuse me.
I am a bit confused about dual space unit vectors in the case of non-orthonormal vector spaces.
(Reference: A Student's Guide to Vectors and Tensors, by Daniel Fleisch, Cambridge, 2012.)
I will be grateful for being corrected in the following:
Suppose V is a non-orthonormal vector space, and V* its dual space. Suppose e1 is in V and e1 its corresponding dual unit vector in V*.
One of the consequences, if I understand it correctly, of the Riesz, or Riesz-Frechet, Representation Theorem is that to e1
there corresponds a u in V such that for every w in V, e1(w) = the inner product of u and w.
Also, starting from e1 in V, e1 is its dual.
The natural conclusion is that u=e1 . Yet the above reference seems to say that they are not.
[1] Let w = another unit vector in V, one which is neither equal nor orthogonal to e1. That is, the inner product of e1 and w is nonzero. Yet on the other side, the above reference states (p 114) that eiej=δij.
[2] In the drawings, the author draws (e.g., pp 117, 119) both vectors and their duals in the same plane,which leads me to think that he is drawing not the actual duals, but the vectors in V which correspond to the duals. But he draws (the vector in V which, by Riesz, corresponds to) ei as being different to ei.
Thanks in advance for showing me where my understanding went awry.
(Reference: A Student's Guide to Vectors and Tensors, by Daniel Fleisch, Cambridge, 2012.)
I will be grateful for being corrected in the following:
Suppose V is a non-orthonormal vector space, and V* its dual space. Suppose e1 is in V and e1 its corresponding dual unit vector in V*.
One of the consequences, if I understand it correctly, of the Riesz, or Riesz-Frechet, Representation Theorem is that to e1
there corresponds a u in V such that for every w in V, e1(w) = the inner product of u and w.
Also, starting from e1 in V, e1 is its dual.
The natural conclusion is that u=e1 . Yet the above reference seems to say that they are not.
[1] Let w = another unit vector in V, one which is neither equal nor orthogonal to e1. That is, the inner product of e1 and w is nonzero. Yet on the other side, the above reference states (p 114) that eiej=δij.
[2] In the drawings, the author draws (e.g., pp 117, 119) both vectors and their duals in the same plane,which leads me to think that he is drawing not the actual duals, but the vectors in V which correspond to the duals. But he draws (the vector in V which, by Riesz, corresponds to) ei as being different to ei.
Thanks in advance for showing me where my understanding went awry.
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