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cbarker1

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MHB

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- Homework Statement
- Show that for any linear functional ##l## on ##B## is continuous if and only if ##A=\{f\in\B:l(f)=0\}## is closed.

- Relevant Equations
- #B# is a Banach space over the complex field and #ker(l)={f\in \mcalB: l(f)=0}#

Dear everybody,

I am having some trouble proving the implication (or the forward direction.) Here is my work:

Suppose that we have an arbitrary linear functional ##l## on a Banach Space ##B## is continuous. Since ##l## is continuous linear functional on B, in other words, we want show that ##l^{-1}\{0\}=A## and this is closed. I am having trouble with this claim.

Thanks

Carter

I am having some trouble proving the implication (or the forward direction.) Here is my work:

Suppose that we have an arbitrary linear functional ##l## on a Banach Space ##B## is continuous. Since ##l## is continuous linear functional on B, in other words, we want show that ##l^{-1}\{0\}=A## and this is closed. I am having trouble with this claim.

Thanks

Carter

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