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Math Help - Continuous Linear Operators

  1. #1
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    Continuous Linear Operators

    Dear Colleagues,

    Could you please help in solving the following problem:

    Let (T)\longrightarrow Y " alt="T(T)\longrightarrow Y " /> be a linear operator, where D(T)\subset X and X, Y are normed spaces. Then:
    If T is continuous at a single point x_{0}\in X with x_{0}\neq0, then it is continuous at every point of X. Here D(T) denotes the domain of T.

    Remark: We want to prove this proposition without using the boundedness of the operator.


    Regards,

    Raed.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by raed View Post
    Dear Colleagues,

    Could you please help in solving the following problem:

    Let (T)\longrightarrow Y " alt="T(T)\longrightarrow Y " /> be a linear operator, where D(T)\subset X and X, Y are normed spaces. Then:
    If T is continuous at a single point x_{0}\in X with x_{0}\neq0, then it is continuous at every point of X. Here D(T) denotes the domain of T.

    Remark: We want to prove this proposition without using the boundedness of the operator.


    Regards,

    Raed.
    What have you done as of yet? The idea is that for any x\in X \displaystyle \lim_{z\to x}T(z)=\lim_{z\to x_0}T(x_0+x-z)=T(x_0)+T(x)-\lim_{z\to x_0}T(z)=T(x). Formalize that.
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