let $X$ be a normed space .let $E$ be a maximal closed subspace of $X$.there exist a bounded linear functional $f$ on $X$ such that ker$(f)=E$

Question

let $X$ be a normed space over field $\mathbb{C}$,let $E$ be a maximal closed subspace of $X$.prove that there exist a bounded linear functional $f$ on $X$ such that ker$(f)=E$

i know that kernal of any functional $f$ is either closed subspace of $X$ or a dense subspace of $X$. and it is closed subspace if and only if $f$ is bounded.

let $x_0\in X/E$, then $E\subset \overline{E+<x_0>}=X$,since $E$ is maximal closed subspace.

from this can i claim that every vector $x$ in $X$ can be written uniquely as $x=p+\alpha\ x_0,$ where $p\in E,\alpha\in\mathbb{C}$

Not directly, you need to show that $E + \langle x_0\rangle$ is a closed subspace to prove the existence. — Daniel Fischer, Aug 27 '17 at 14:06

score 1 · Accepted Answer · answered Aug 27 '17 at 14:10

You can consider $Y=X/E$ with the quotient norm; by assumption, it is a non trivial topological vector space with no closed subspace except for $\{0\}$ and $Y$. Now note that if $y\in Y$, $y\ne0$, then $\langle y\rangle$ is topologically isomorphic to $\mathbb{C}$, hence closed.

Can you finish?

let $X$ be a normed space .let $E$ be a maximal closed subspace of $X$.there exist a bounded linear functional $f$ on $X$ such that ker$(f)=E$

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