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Math Help - Proof for banach Space 2

  1. #1
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    Proof for banach Space 2

    Prove that, In a normed linear space Xn converges to X and Yn converges to Y then XnYn converges to XY .

    Xn and Yn are sequence in banach space.

    Anyone idea ?
    Last edited by neset44; May 10th 2010 at 01:35 AM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by neset44 View Post
    In a normed linear space Xn converges to X and Yn converges to Y then Xn.Yn converges to X.Y

    Anyone idea ?
    What is X_n.Y_n?
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  3. #3
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    sequence in banach space
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  4. #4
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    Quote Originally Posted by Drexel28 View Post
    What is X_n.Y_n?
    Quote Originally Posted by neset44 View Post
    sequence in banach space
    What Drexel28 is asking is: What does the dot mean in X_n.Y_n? It looks as though it is supposed to be an inner product. But a Banach space doesn't have an inner product (unless it is a Hilbert space), so it's hard to know what this problem is supposed to be about.
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  5. #5
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    In fact, although you titled this "proof for Banach space 2", you only said "In a normed linear space Xn". Not all normed spaces are Banach spaces. Please state the problem in full.
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  6. #6
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    Xn and Yn are in Banach space. So in a normed linear space Xn converges to X and Yn converges to Y then prove that XnYn (multiply) converges to XY.

    That is the full problem.
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  7. #7
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    Quote Originally Posted by neset44 View Post
    Xn and Yn are in Banach space. So in a normed linear space Xn converges to X and Yn converges to Y then prove that XnYn (multiply) converges to XY.

    That is the full problem.
    There is no operation of multiplication defined in a Banach space, so this question makes no sense.

    In order for a product X_n.Y_n to be defined, there must be some additional structure in the space. It has to be either a Hilbert space (with an inner product), or a Banach algebra (with a multiplicative structure).

    Also, as HallsofIvy has pointed out, a Banach space is not the same as a normed linear space.
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  8. #8
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    i changed product with standart multiply as you see. but anyway;

    This is the full problem .

    It says on theorem :

    In a normed linear space Xn coverges to X and Yn converges to Y => XnYn converges to XY . Prove it ?

    Nothing more
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  9. #9
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    One more time and then we give up!

    A "Banach space" is a vector space having a norm such that "Cauchy sequences" converge.

    While "scalar multiplication" and "vector addition" are defined in any vector space, multiplication of vectors is NOT generally defined. There is no "standard multiply"! It is impossible to answer your question with knowing how you are defining "XnYn".
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  10. #10
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    i asked and My teacher said . It is a scalar multiplication
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  11. #11
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    Are you saying that {Xn} is a sequence of scalars that converge to X and {Yn} is a sequence of vectors in a Banach space that converge to Y? That was not what you said initially!
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  12. #12
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    as a result that what he given to me Xn converges to x and Yn converges to y then XnYn converges to xy ?

    there is not any more information. then he said between Xn and Yn multiplication is scalar ...

    that is :S
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  13. #13
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    Quote Originally Posted by neset44 View Post
    as a result that what he given to me Xn converges to x and Yn converges to y then XnYn converges to xy ?

    there is not any more information. then he said between Xn and Yn multiplication is scalar ...

    that is :S
    You really need to improve you english writing skills.

    That aside, if you meant (x_n) \subset \mathbb{C} (or \mathbb{R} it works the same) and (y_n) \subset X (where this last one is a normed space over whichever field you pick from the above) such that |x_n - x| \rightarrow 0 and \| y_n - y \| \rightarrow 0 then

    \| x_ny_n - xy \| \leq \| x_ny_n -x_ny\| + \|x_ny-xy\| \leq M\| y_n-y\| + |x_n-x| \|y\| \rightarrow 0

    where |x_n|\leq M for all n.
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  14. #14
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    My guess is that he is talking about a Banach Algebra.
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