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If $X$ is a banach space and $M\subset X $

Define $M^{\perp} = \{x'\ :\ x'(x)=0\ \forall x\in M\}$ and $N^\perp = \{x\in X\ :\ x'(x)=0\ \forall x' \in N\}$ where $x'$ denotes the functional from the dual.

If $X$ is a normed space and $U$ is a closed suset of $X$, how do I show that there exist canonical isometric isomorphism, i.e. $(X/U)' \cong U^\perp $ and

$U'\cong X'/U^\perp$

I need help to learn this problem. Thanks.

1 Answers 1

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How do elements of $U^\perp$ act as functionals on $X$?

Can you extend this to make them act as a functionals on $X/U$ in a consistent way? I guess this means: if $x+U=y+U$, you have to make sure that each of your functionals all give these the same value.

This should give you a candidate definition for your first isomorphism; then check that it is really an isometric isomorphism by chasing definitions.

Similar sorts of things will get you the answer to the second exercise.