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show/hide this revision's text 2 tweaked the formatting and corrected a couple of typos

Hi everybody, I've got an exercise about Banach spaces and I can't see how to solve it. It is a very simple problem and I know it might be some little detail I'm missing, and that is why I'm asking for help.

It says:

Let X be a Banach space with a monotone basis. Let $σ$ be the set of all finite block basis bases in the unitary unit ball of X that contain at least one vector xi of norm 1. Suppose (y'1, z'1, y'2, z'2,...,y'n, z'n) is in $σ$. Prove that the norms ||$Σ$i=1n$\Vert \sum_{i=1}^n (y'iy'_i + z'i)|| z'_i)\Vert$ and ||$Σ$i=1n$\Vert \sum_{i=1}^n (y'iy_i' - z'i)|| z'_i)\Vert $ are both at least 1/2.

Any help is welcome.

Thanks.
show/hide this revision's text 1

Problem in Banach space

Hi everybody, I've got an exercise about Banach spaces and I can't see how to solve it. It is a very simple problem and I know it might be some little detail I'm missing, and that is why I'm asking for help.

It says:

Let X be a Banach space with a monotone basis. Let $σ$ be the set of all finite block basis in the unitary ball of X that contain at least one vector xi of norm 1. Suppose (y'1, z'1, y'2, z'2,...,y'n, z'n) is in $σ$. Prove that the norms ||$Σ$i=1n(y'i + z'i)|| and ||$Σ$i=1n(y'i - z'i)|| are both at least 1/2.

Any help is welcome.

Thanks.