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In the paper "Commutators on $\ell_\infty$" by Dosev and Johnson, in Lemma 4.2 Cas II, the authors have said that "There exists a normalized bock basis $\{u_i\}$ of $\{x_i\}$ and a normalized block basis $\{v_i\}$ of $\{y_i\}$ such that $\|u_i-v_i\|<\frac{1}{i}.$ Does anyone have any idea to prove this?

More elaborately, we have two subspaces $X$ and $Y$ of $\ell_\infty$ both isomorphic to $c_0.$ $\{x_i\}$ and $\{y_i\}$ are bases of $X$ and $Y$ respectively which are equivalent to standard base of $c_0$. We have also $X\cap Y=\{0\}$ and $d(X,Y):=\inf\{\|x-y\|:x\in X,y\in Y, \|x\|=1\}=0.$ Now how to show "There exists a normalized bock basis $\{u_i\}$ of $\{x_i\}$ and a normalized block basis $\{v_i\}$ of $\{y_i\}$ such that $\|u_i-v_i\|<\frac{1}{i}.$" ?

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    $\begingroup$ Hint: Check that for each $n$, $d(X_n,Y_n) =0$, where $X_n$ is the linear span of $x_n, x_{n+1}, \dots$ and similarly for $Y_n$. $\endgroup$
    – Bill Johnson
    Commented Jun 23, 2019 at 14:34
  • $\begingroup$ @ Bill. Thank you for the hint. I am writing the details just to complete the argument. Following Bill, we have $X=X_n\oplus \tilde{X}_n$, $Y=Y_n\oplus \tilde{Y}_n$ (As dim $\tilde{X}_n$ & $\tilde{Y}_n$ finite, this is ok.) Let $\|\alpha_k-\beta_k\|\to 0,$ $\alpha_k\in X,\beta_k\in Y$ and $\alpha_k=t_k+s_k$ and $\beta_k=r_k+l_k$ chosen according to the decomposition. Since $\tilde{X}_n+\tilde{Y}_n$ is complemented in $X+Y$ we clearly have that $\|s_k-l_k\|\to 0.$ This establishes Bill's claim. Now choosing block basis is easy since we have freedom to cut off the series in $x_n$'s and $y_n$'s. $\endgroup$
    – A beginner mathmatician
    Commented Jun 25, 2019 at 6:53

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