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Thread: strong topology

  1. #1
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    strong topology

    Hi all,
    i have in my lecture notes this statement :
    " the strong topology is equivalent to the topology given by the norm ||w||=sup |(w|v)|
    which makes the E' (dual of E ) a Banach space , and the space E is the subspace of E''
    "

    why the strong topology is equal to this norm topology ?
    and why the space E is the subspace of E'' ?
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  2. #2
    Super Member girdav's Avatar
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    $\displaystyle E$ is not "really" a subspace of $\displaystyle E''$ . It is the case after an identification. We define $\displaystyle J : E\rightarrow E''$ by the following way. Let $\displaystyle \varphi_x$ defined by $\displaystyle \varphi_x(f):= f(x)$. $\displaystyle \varphi_x\in (E')'$ and $\displaystyle J(x) =\varphi_x$ makes $\displaystyle J$ linear.
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  3. #3
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    thanks , i dont get u mean , is it possible to clrify more ?
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  4. #4
    Super Member girdav's Avatar
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    $\displaystyle J(E)$ is a subspace of $\displaystyle E''$. We identify $\displaystyle E$ with a subspace of $\displaystyle E''$.
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  5. #5
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    thanks, but when i see my lecture notes again i saw same experssion that :
    " because A is a subspace of A'' ,the elements a belongs to the A can be viewed as functions on X(A) ( characters of A ) by setting :
    a(x):=X(a)
    "
    and still i am confusing about the double dual , how space is the subspace og that , and also i can't understand the meaninig of this phrase that i wrote .
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