# Application of Hahn-Banach Theorem

Let $E$ be Banach space over $\mathbb{R}$. Let $u$ and $v$ be such that $||u||=||v||=1$ and $||2u+v||=||u-2v||=3$.

How do we show that there exists a linear functional $f$ defined on all of $E$ such that $||f||=1$, $f(u)=1$ and $f(v)=1$?

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Please do not write in caps. Most of the time it is considered to be very rude to do so. Also, try to show us what you have tried before, so we can help you solve the problem, instead of giving you the solution on a silver platter. This way you will learn a lot more! – sxd Aug 23 '12 at 23:15
I think you need to show first that $u$ and $v$ are linearly independent, so that they span a 2-dimensional subspace. – timur Aug 24 '12 at 0:06
I defined the functional: g defined in H=span{u,v} such that, g(au+bv)=a+b. Note that g(u)=g(v)=1 end g(2u/3+v/3)=1. But I can't show that ||g||=1. – Valdinês Júnior Aug 24 '12 at 0:13
@DimitriSurinx I find it ridiculous to claim that typing in all caps (in this case) is rude. – Quinn Culver Aug 24 '12 at 0:31

Note first that $u$ and $v$ are linearly independent. Now consider the linear functional $g$ on the linear span $V$ of $u$ and $v$ such that $g(u) = 1$ and $g(v) = 1$. Thus $g(au + bv) = a+b$. What is the norm of this? Draw a picture of the unit ball of $V$, using the fact that it must be symmetric and convex...