BMO and the Banach Space Approximation Problem
Copyright policy )BMO and the Banach Space Approximation Problem
Let \(L^{\infty}=L^{\infty}(\partial D)\), \(H^{\infty}=H^{\infty}(D)\), \(BMO(\partial D)=the\) space of functions f on \(\partial D\) with \(\int^{2\pi}_{0}f(t)dt=0\) and \(\| f\|_{BMO}=\sup \{(\frac{1}{| I|}\int_{I}| f-f_ I|^ 2dt)^{1/2}:\) I an \(arc\subset \partial D\}0\), any \(x_ 1,...,x_ n\) in X there is a linear operator T:X\(\to X\) such that \(\| x_ j-Tx_ j\| \leq d\| x_ j\|\) \((j=1,...,n)\), \(\| T\| \leq A\) and rank \(T\leq C(n,d)\), where A is independent of \(d,x_ 1,...,x_ n\). By Heinrich's theorem, X has u.a.p. iff \(X^*\) has u.a.p. The author proves two theorems implying that BMO (and hence each of \(H^ 1_ 0\), \(H^ 1\), BMO(\(\partial D)\), \(L^{\infty}/H^{\infty}\cong BMOA)\) has the u.a.p. The proofs are involved and heavily combinatorial.
uniform approximation property, bounded-mean-oscillation function spaces, Geometry and structure of normed linear spaces, Abstract approximation theory (approximation in normed linear spaces and other abstract spaces), analytic Poisson extension, Fefferman's duality theorem, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Banach spaces of continuous, differentiable or analytic functions
uniform approximation property, bounded-mean-oscillation function spaces, Geometry and structure of normed linear spaces, Abstract approximation theory (approximation in normed linear spaces and other abstract spaces), analytic Poisson extension, Fefferman's duality theorem, Banach algebras of differentiable or analytic functions, \(H^p\)-spaces, Banach spaces of continuous, differentiable or analytic functions
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