Definition talk:Linear Functional

Suggest replacing $\mathbb F \in \{ \R,\C \}$ with $\mathbb F$ any field. Since the link Definition:Vector Space and Definition:Vector Subspace are given for an arbitrary field, this isn't superseding the present accessibility of the definition. --Linus44 (talk) 13:01, 5 March 2013 (UTC)


 * Yes, seems fine to me. We will care about adding comments about fields of numerals being most usual when the foundations of complex and (probably) multidimensional analysis are laid down enough to support a proper covering of functional analysis. &mdash; Lord_Farin (talk) 13:05, 5 March 2013 (UTC)


 * ok. D'ya mind if I edit Equivalence of Definitions of Norm of Linear Functional? Was going to add a proof but I think it'll be a tidier proof with a different notation for all the norms $\Vert L \Vert$ --Linus44 (talk) 13:16, 5 March 2013 (UTC)


 * What is your anticipated change? The notation was all taken from Conway. If it just involves subscripts to indicate which norms are from $H$ and which pertain to functionals, I don't see a problem (note though that Hilbert spaces are defined now only over $\R$ or $\C$; perhaps this can be changed into fields that are $\R$-algebras but I don't have sources treating such generalities).


 * FFR please disregard any of my User Stubs on pages with Conway as source, I haven't touched those in at least a year... Got bogged down by the lack of complex analysis coverage. &mdash; Lord_Farin (talk) 13:33, 5 March 2013 (UTC)


 * I just meant if the norms labelled $(1)$--$(4)$ were $\Vert L \Vert_1$--$\Vert L \Vert_4$ then writing out all the necessary inequalities $\Vert L \Vert_2 \leq \Vert L \Vert_1 \leq \Vert L \Vert_3 \leq \Vert L \Vert_2$...etc would be clearer in the proof than writing out the definitions each time. --Linus44 (talk) 13:42, 5 March 2013 (UTC)


 * Re: scope of definition of Hilbert spaces, I don't think it's so important in that case: as far as I know, Hilbert spaces are only studied over $\R,\C$ and maybe over $p$-adic fields (the last case being way distant if it is interesting). Whereas linear functionals show up all over the place, so having any field here is less likely to end up with pages linking back to a not-general-enough definition. --Linus44 (talk) 13:46, 5 March 2013 (UTC)


 * To the best of my knowledge you are right (though I have seen advances in the direction of quaternion Hilbert spaces). With all of my comments and remarks out of the way, feel free to go ahead :). &mdash; Lord_Farin (talk) 13:54, 5 March 2013 (UTC)