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Math Help - Why does this proof imply 'continuous'?

  1. #1
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    Why does this proof imply 'continuous'?

    Hi!

    The map \langle ; \rangle : X \times X \to \mathbb{K} (it is a dot product) is continuous.

    Proof:

    |\langle x,y \rangle - \langle x' , y' \rangle|

    = |\langle x-x' , y \rangle - \langle x' , y'-y \rangle |

    \le |\langle x-x' , y \rangle| - |\langle x' , y'-y \rangle |

    \le ||y||\cdot ||x-x'|| + ||x'||\cdot ||y'-y||


    So what is the argument to imply that the dot product continuous?

    Any help would be much appreciated. Thank you!

    Rapha
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  2. #2
    Moo
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    Hello,

    To prove that <,> is continuous, you just have to check that \forall (x,y)\in X^2, there exists M such that \langle x,y\rangle\leq M \cdot \|x\|_X\cdot\|y\|_X

    And by the Cauchy-Schwarz inequality, this is obvious :
    Spoiler:
    M=1
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  3. #3
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    Hi Moo.

    Alright then

    Thank you very much!

    best regards
    Rapha
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