3 ms·
You are right about infinite dimensions, wrong about finite dimensions. V and V* are naturally isomorphic for finite dimensions. In finite dimensions, V and V*
by btilly 2y ago
You are right about infinite dimensions, wrong about finite dimensions. V and V* are naturally isomorphic for finite dimensions.
In finite dimensions, V and V* are isomorphic, but not naturally so. The isomorphism requires additional information. You can specify a basis to get the isomorphism, but many bases will give the same isomorphism. The exact amount of information that you need is a metric. If you have a metric, then every orthonormal basis in that metric will give the same isomorphism.
- ogogmad 2y agoYou need to correct the 2nd sentence to say that V and V** are naturally isomorphic. V and V* are only unnaturally isomorphic. All of this holds only in finite dimensions, of course.
- g15jv2dp 2y agoYou have a typo in your first line, and I answered a sibling comment about that. Metrics are irrelevant to the discussion (and I presume you wanted to write "norm" instead of "metric").
- btilly 2y agoYes, I had a typo. As for why I said metric, see https://en.wikipedia.org/wiki/Metric_tensor https://en.wikipedia.org/wiki/Metric_tensor. Which is technically a concept from differential geometry rather than linear algebra. But then again, tensors are literally the topic that started this. And it is only in differential geometry that I've ever cared about mapping from V to V*.
- g15jv2dp 2y agoThere are no manifolds at all in this discussion... What are you even talking about? Just stringing words you vaguely know together??
- btilly 2y agoThis comment is in a discussion about an article titled, Tensors, the geometric tool that solved Einstein's relativity problem. Therefore, "tensors are literally the topic that started this discussion." Hopefully that's a hint that you should attempt to figure out what someone might be talking about before going to schoolyard insults.
- g15jv2dp 2y agoSchoolyard insult? Uh? Can you quote the part of my comment that would be the "schoolyard insult"? > This comment is in a discussion about an article titled, Tensors, the geometric tool that solved Einstein's relativity problem. Therefore, "tensors are literally the topic that started this discussion." Again, are manifold involved in any way in the definition of tensors and their properties? No? Then why are you even mentioning "metric tensors"? (Which aren't even tensors, but tensor fields...)