let $ E $ be a normed space that can be represented as a direct sum of two vector subspaces: $ E = F + G $. Show that $\frac{ E}{ F}$ is isometric with $ G $ Using the quotient norm since $ F $ is a closed subspace I want to build the bijective mapping between the quotient space and$ G$. One idea is to take an ismorphism theorem that is an isometry but I have not gotten far. The other is to directly construct the function but I don't see how to proceed from the definition of quotient norm, maybe it is some extension. Will you have any comments or suggestions, thanks.
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$(1).\ $ the projection $\pi:E\oplus F\to E$ is linear and has kernel equal to $F$, so the first isomorphism theorem applies to say.....
$(2).\ \pi $ maps the unit ball $B^{E\oplus F}_1(0)$ onto the unit ball $B^{E}_1(0)$, so...
Matematleta
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By $E\otimes F$ perhaps do you mean $E=F\oplus G$? – Berci Oct 26 '19 at 18:50
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Yes, it's a typo. Thank you. – Matematleta Oct 26 '19 at 18:51