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Here is a result from Hahn-Banach Theorem:

Let $V$ be a normed vector space and $W$ be a closed subspace of $V$. Let $x\in V\setminus{W}$, then there exists $f\in V^*$ such that $f(x)=\|x\|$, $\|f\|=1$ and $f(w)=0$ for all $w\in W$.

Here is my attempt to prove this result:

Since $x\in V$ and $W$ is a closed sunspace of $V$ then let $x+W\in V/W$ and $V/W$ is the quotient space. Now using a corollary from Hahn-Banach theorem on our quotient space "If $x\in V$ then $\exists$ $f\in V^*$ with $f(x)=\|x\|$ and $\|f\|=1$. " Hence, there exists a linear functional $f_{0}$ on $V/W$ such that $\|f_{0}\|=1$ and $f_{0}|_{V}=\|x\|=f(x)$ with $f\in V^*$ since $x\in V$ and $f_{0}|_{W}=0$ since $x\not\in W$ and $\|f\|=1$.

Iam not sure about this proof and I would appreciate any help..

S.N.A
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1 Answers1

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The subspace $N=Vect(x)+W$ is closed see

Sum of closed subspaces of normed linear space

Consider the linear function defined on $N$ by $f(x)=\|x\|, f(m)=0, w\in W$, it is continuous since its kernel $M$ is closed, by Hahn Banach, you can extend it to $V$ in $g$.

Tsemo Aristide
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