4 ms·
Remember that V* * = V is valid only for vector spaces with finite dimensions.
by Lucasoato 6y ago
Remember that V* * = V is valid only for vector spaces with finite dimensions.
- pmiller2 6y agoYep! The simplest explanation for why this is true is that when V is infinite dimensional, V^* and V^{* *} need not have the same dimension, which precludes them being isomorphic.
- phreeza 6y agoInteresting, can this happen for function spaces?
- klodolph 6y agoMany function spaces are infinite dimensional.
- phreeza 6y agoI know, but I was trying to think of a space where the dual of the dual is not the primal? I think in Hilbert spaces this does not happen?
- klodolph 6y agoYou're thinking in terms of additional structure. In other words, you're no longer talking about dual vector spaces, but something different.
- symplectico 6y agoThat is not exactly true. For example every Hilbert space is reflexive, even if infinite dimensional.
- RossBencina 6y agoIt is true that finite dimensional vector spaces are reflexive. But the broader claim that only finite dimensional vector spaces are reflexive is false. For example, Hilbert spaces are reflexive and infinite dimensional. See the last paragraph here: https://en.wikipedia.org/wiki/Hilbert_space#Duality https://en.wikipedia.org/wiki/Hilbert_space#Duality See also: https://en.wikipedia.org/wiki/Reflexive_space https://en.wikipedia.org/wiki/Reflexive_space
- edflsafoiewq 6y agoOP is correct assuming "dual" is defined as all the linear functionals on a space. The dual of an infinite-dimensional space in this sense always has larger dimension than the original space. The difference comes in that for Hilbert spaces, "dual" is usually taken to mean only the continuous linear functionals.
- snicker7 6y agoeg "continuous" vs "algebraic" duals.