Let ( X , ‖ ⋅ ‖ ) be a normed space Define r = sup { ‖ x 1 − x 2 ‖ : x 1 , x 2 ∈ X , ‖ x 1 + x 2 ‖ = 2 , ‖ x 1 ‖ = ‖ x 2 ‖ = 1 } .Question: Is r = 0 ?

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Let   (  X  ,  ‖  ⋅  ‖  ) be a normed space Define  r  =  sup  {  ‖      x    1    −      x    2    ‖  :      x    1    ,      x    2    ∈  X  ,  ‖      x    1    +      x    2    ‖  =  2  ,  ‖      x    1    ‖  =  ‖      x    2    ‖  =  1  }  .Question: Is   r  =  0  ?

Savanah Hernandez · Accepted Answer

Step 1Take for example   X  =      ℓ          ∞       or       c    0   with the supremum norm, and take       x    1   to be the sequence   {      1    n        }          n      =      1              ∞       and       x    2   the sequence   {      1          n      2            }          n      =      1              ∞       . Then   ‖      x    1    +      x    2    ‖  =  2 , and   ‖      x    1    ‖  =  ‖      x    2    ‖  =  1 , but  ‖      x    1    −      x    2    ‖  =      sup          n            |        1    n    −      1          n      2            |    ≥      1    2    −      1    4    =      1    4  so   r  ≥      1    4   .

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