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very clearly that this is meant to be the purpose. More importantly, though, as a physicist and not a mathematician, some of the details are unclear to me, and I think need to be clarified for a less specialized audience. For example, does the space of continuous functions need to include the dirac-delta function, since it is only the limit of a series of functionals and not a regular function, per-se? If this is the only counter example to the non-degeneracy, does that actually prove anything? Second of all, what is meant when it says that it is in the space of the dual but "not of the required form"? What is the required form to have the dirac delta function as its dual? This is very confusing to me.
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This section is extremely unclear. First of all, I gather that it's meant to be an example of how in infinite dimensions a bilinear form can satisfy f(x,y)=0 for all y only if x=0 without being an isomorphism, and therefore doesn't satisfy the criteria for a non-degenerate form, but it doesn't state
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It would be much more useful if the article starts with describing the Non-degenerate forms rather than degenerate forms. Non-degenerate forms are much more used and more general. Degeneracy in mathematics is not very important. If nobody objects can we change the title and corresponding article to
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What is the significance of this "if and only if"? The thought behind it seems to be "A symmetric bilinear form is nondegenerate if and only if..." Perhaps it's a relic of a time when the paragraph began like that. Should it be deleted?
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therefore that it is meant to be the continuous dual. But then the article could use a hint that "degeneracy" may depend on the topology of V that one is working with. –
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the notation V* refers to? It it's the algebraic dual, then all forms are degenerate because V* is too large to be isomorphic to V.
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This is very confusing and unclear. If anyone wants to sort it out, the same problem is present on the Inner product space page.
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The
Examples section just repeats the last paragraph of Nondegenerate forms. Maybe some concrete examples could be added?
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are duals ("A nondegenerate form is one that is not degenerate") and therefore mostly redundant. Given that
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on 6 January 2021. For the contribution history and old versions of the redirected page, please see
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Knowledge. If you would like to participate, please visit the project page, where you can join
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It's not a minus sign, it's a placeholder for an omitted argument.
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Proposed merge of
Nondegenerate form with Degenerate bilinear form
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is quite short it makes sense to merge these articles.
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No, it's ok. The phrase continues on the next line … --
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or equivalently in finite dimensions, if and only if
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