= Small perturbation of a complemented block basis
Suppose $(u_n)$ is a normalized <block basic sequence> in the <l-p sequence space>, $1\le p<\infty$, and $\sum_n\|x_n-u_n\|_p<1$. Choose norm-one <linear functionals> $\phi_n$ supported on the respective blocks with $\phi_n(u_n)=1$. The <bounded linear operator> $A=I+\sum_n\phi_n(\cdot)(x_n-u_n)$ is invertible by the <Neumann series>. The block projection $Qz=\sum_n\phi_n(z)u_n$ has norm at most one, and $AQA^{-1}$ projects onto the closed span of $(x_n)$. Thus that span is a <complemented subspace> isomorphic to the <l-p sequence space>.
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