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Thread: Group theory problem 2

  1. #1 Group theory problem 2 
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    Ok, we've warmed up enough! Now for the real thing:

    Let and be topological groups, being connected as a topological space, and let be a homomorphism of topological groups, i.e.: Continuous, and: (or , in simplified notation) . Suppose moreover that , i.e.: such that: . Then , i.e.: is onto.

    Can anyone suggest a proof? Then I will explain why I picked this problem up.


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  3. #2 Re: Group theory problem 2 
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    Quote Originally Posted by Obelix
    Ok, we've warmed up enough! Now for the real thing:

    Let and be topological groups, being connected as a topological space, and let be a homomorphism of topological groups, i.e.: Continuous, and: (or , in simplified notation) . Suppose moreover that , i.e.: such that: . Then , i.e.: is onto.

    Can anyone suggest a proof? Then I will explain why I picked this problem up.
    This is rather obvious.

    Since you have a predilection for posting trivial problems and have yet to produce anything resembling an elegant or insightful argument, you have an opportunity to redeem yourself. Produce one -- say six lines or less.

    No cheating by copying a solution from any of the other forums in which you have posted the same problem.


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  4. #3 Re: Group theory problem 2 
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    It is pretty clear now, given that a couple of days have passed in silence, that Obelix is not up to providing his own proof.

    So, for those who might find this problem a bit interesting here is a solution. It assumes some elementary facts regarding topological groups.

    Proof: Since has non-empty interior and since is continuous homomrphism we can, by translation if necessary, assume that contains a neighborhood of the identity. Since it is a group homomorphism contains the subgroup of generated by that neighborhood. But a connected group is generated by any neighborhood of the identity. QED


    For anyone interested in pursuing the topic of topological groups I recommend the classic treatise by L.S. Pontryagin Topological Groups. For any electrical engineering control theory types, this is the Pontryagin of "the Pontryagin maximum principle". For any interested in harmonic analysis, this is the Pontryagin of "Pontryagin duality". For those interested in vector bundles this is the Pontryagin of "Pontryagin classes". He did rather well, as a mathematician at least (not necessarily as a nice guy), for someone who was blinded at age 14.
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  5. #4  
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    Just as a matter of interest. Every Lie group is a topological group, but is the converse true? I suspect not, just as not every top. space is a manifold.
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  6. #5  
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    Quote Originally Posted by Guitarist
    Just as a matter of interest. Every Lie group is a topological group, but is the converse true? I suspect not, just as not every top. space is a manifold.
    Lie Groups are a proper subclass of topological groups.

    The topological vector spaces that you are studying in functional analysis are topological groups with additional structure. Infinite-dimensional topological vector spaces are not locally compact, while Lie groups are, since they are finite-dimensional manifolds.

    In the case of a Lie group, the group operations are not only continuous, they are smooth.
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  7. #6  
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    Quote Originally Posted by Guitarist
    Just as a matter of interest. Every Lie group is a topological group, but is the converse true? I suspect not, just as not every top. space is a manifold.
    Lie Groups are a proper subclass of topological groups.

    The topological vector spaces that you are studying in functional analysis are topological groups with additional structure. Infinite-dimensional topological vector spaces are not locally compact, while Lie groups are, since they are finite-dimensional manifolds.

    In the case of a Lie group, the group operations are not only continuous, they are smooth.
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  8. #7  
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    Quote Originally Posted by DrRocket
    The topological vector spaces that you are studying in functional analysis are topological groups with additional structure. Infinite-dimensional topological vector spaces are not locally compact, while Lie groups are, since they are finite-dimensional manifolds.
    Not that I'm ready to tackle functional analysis, but just out of curiosity based on my limited knowledge, are Banach and Hilbert spaces locally compact?
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  9. #8  
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    Quote Originally Posted by AlexP
    Quote Originally Posted by DrRocket
    The topological vector spaces that you are studying in functional analysis are topological groups with additional structure. Infinite-dimensional topological vector spaces are not locally compact, while Lie groups are, since they are finite-dimensional manifolds.
    Not that I'm ready to tackle functional analysis, but just out of curiosity based on my limited knowledge, are Banach and Hilbert spaces locally compact?
    All locally compact topological vector spaces are finite dimensional. All finite dimensional topological vector spaces of a given fixed dimension are isomorphic in the category of topological vector spaces.
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  10. #9  
    Moderator Moderator AlexP's Avatar
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    I asked because I wondered if it had anything to do with being complete. Just a guess. Is is true that a topological vector space is locally compact if and only if it is finite-dimensional?
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  11. #10  
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    Quote Originally Posted by AlexP
    I asked because I wondered if it had anything to do with being complete. Just a guess. Is is true that a topological vector space is locally compact if and only if it is finite-dimensional?
    What I stated immediately implies that a topological vector space is locally compact if and only if it is finite dimensional.

    There are lots of complete infinite-dimensional spaces.
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