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Could someonoe help me to decide if the following satetement is true?

If $K$ is a strictly convex Banach space and $a,b\in{K}$ verify $|a-b|=|a|-|b|=1$ then, $a=\lambda{b}$ for some $\lambda\geq{0}$.

I only need the case $K=\mathbb{R}$ and $K=\mathbb{C}$ but I wonder if there exists a proof in a general strictly convex banach space.

Thanks.

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  • $\begingroup$ The third property for strictly convex spaces listed on the Wikipedia article is that strict convexity of a space is equivalent to $\|x+y\|=\|x\|+\|y\|$ implies $x=\lambda y$ for some $\lambda>0$. Your claim follows from this, using the substitution $x=a-b$ and $y=b$. $\endgroup$ – David M. Apr 14 at 14:55
  • $\begingroup$ My definition of strictly convex banach space is that $\forall{x,y}$ with $||x||=||y||=1$, and $||x+y||=2$, then $x=y$. But I don't see how this definition implies Wikipedia's definition $\endgroup$ – mathlife Apr 14 at 15:44

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