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General: Request for reopening "infinitely many linear equations in infinitey many variables"

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    Hi,

    I have a kind request that my community wiki thread "infinitey many linear equations in infinitely many variables" gets reopened. I have editted my original text, so I now hope it is better suitable for MO. Here is a link to it -> http://mathoverflow.net/questions/36348/infinitely-many-linear-equations-in-infinitely-many-variables

    Thanks,

    efq
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    I was the third vote to reopen. There is still room for interpretation of what kind of summation you're interested in, but I think that with the background and motivation provided it is focused enough for a community wiki reference request.
  3.  
    Thank you for your vote! The problem is of course that I can´t be much more specific as I have almost no idea of the subject to start with (e.g. common types of such systems and similar). Among other things, that is also the reason why I think such a question is actually suitable for MO, where the members have much better overview over the fields than on the typical undergraduate/graduate students forums. Or to put it in another way, in this case MO seems to be the only place to get a reasonable and qualified answer to a relatively unreasonable question :-)
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    I too have voted to reopen the question in its new form. I think it's well-intended and hopefully will attract interesting answers.

    • CommentAuthorYemon Choi
    • CommentTimeAug 23rd 2010
     

    I still think that in the present context "infinite dimensional linear algebra" = "linear functional analysis" = "large chunk of Dunford & Schwarz". To be slightly less flippant: if someone asks for "some introductory literature focused on such infinite systems of linear equations in infinitely many unknowns over C" then I am sorely tempted to say "go and read all the classical stuff on the Fredholm alternative, then the spectral theorem for normal operators on Hilbert space, then some of the operator theory on other Banach spaces, then look up K\"othe spaces ..." because without all this, I don't know what can be said in general. This also seems to be what an (under)graduate supervisor would be for.

    On the other hand, if the question is "here is a system of linear equations in infinitely many unknowns, where I have this a priori knowledge about the coefficients of the equations, then do I have some reasonable criteria for inverting the corresponding linear operator", then that seems more like a question which can be answered well.

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    Yemon: Because of the way that the question is worded, I think that many of your remarks might be useful in an answer. I don't see it as asking for a comprehensive answer/encyclopedia article, but rather for some orientation on where to begin doing the OP's own work in learning some of the relevant theory. It is possible that the OP doesn't have a supervisor with as much insight into such things as, say, you or some other MO users.
    • CommentAuthorVP
    • CommentTimeAug 23rd 2010
     

    I agree with Yemon and I also don't understand how can someone study functional analysis for 2 (or even 3?) semesters and not know anything about Ax=b, which is the most fundamental problem in operator theory there is.

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    "how can someone study functional analysis for 2 (or even 3?) semesters and not know anything about Ax=b, which is the most fundamental problem in operator theory there is."

    Could you elaborate on "not know anything about Ax=b" a little bit more? For instance, your observation is based on?
  9.  
    Yemon said it all.
    • CommentAuthorYemon Choi
    • CommentTimeAug 23rd 2010
     

    (... although I forgot to check how to spell Jacob Schwartz's name correctly. Oops.)

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