Skip to main content
MathOverflow

Return to Question

Post Closed as "Duplicate" by Tomasz Kania, leo monsaingeon, LeechLattice, Daniel Loughran, DamienC
Proofreading
Source Link
LSpice
LSpice
  • 12.9k
  • 4
  • 45
  • 69

I am looking for a resource or some hints on why the two normed spaces $\ell_\infty$ and $L_\infty$ belong to the family $\mathcal{B}_1$  , that is,
they are of the family of Banach spaces $X$ such that for any Banach space $X'$ containing $X$ there is a projection $P$ from $X'$ to $X$ such that $\|P\|\leq 1$$\lVert P\rVert\leq 1$.

I am looking for a resource or some hints on why the two normed spaces $\ell_\infty$ and $L_\infty$ belong to the family $\mathcal{B}_1$  , that is,
they are of the family of Banach spaces $X$ such that for any Banach space $X'$ containing $X$ there is a projection $P$ from $X'$ to $X$ such that $\|P\|\leq 1$

I am looking for a resource or some hints on why the two normed spaces $\ell_\infty$ and $L_\infty$ belong to the family $\mathcal{B}_1$, that is,
they are of the family of Banach spaces $X$ such that for any Banach space $X'$ containing $X$ there is a projection $P$ from $X'$ to $X$ such that $\lVert P\rVert\leq 1$.

Source Link
NotaChoice
NotaChoice
  • 141
  • 3

Both $\ell_\infty$ and $L_\infty$ belong to $\mathcal{B}_1$

I am looking for a resource or some hints on why the two normed spaces $\ell_\infty$ and $L_\infty$ belong to the family $\mathcal{B}_1$ , that is,
they are of the family of Banach spaces $X$ such that for any Banach space $X'$ containing $X$ there is a projection $P$ from $X'$ to $X$ such that $\|P\|\leq 1$