Approximation property gives finite-dimensional factorization on compacta

If $K$ is empty use zero maps. Otherwise consider all pairs $(x,\rho )$ with $x\in K$ and $\rho >0$ such that $\mathrm{\parallel }T(z)-T(x){\mathrm{\parallel }}_Y<\epsilon \mathrm{/}4$ whenever $\mathrm{\parallel }z-x{\mathrm{\parallel }}_X<3\rho$. Continuity ensures that the balls $B(x,\rho )$ from this family cover $K$; take a finite subcover $B(x_i,{\rho }_i)$. Put $\delta ={\min }_i{\rho }_i>0$. For $x\in K$ and $z\in X$ with $\mathrm{\parallel }z-x{\mathrm{\parallel }}_X<\delta$, select $i$ with $\mathrm{\parallel }x-x_i{\mathrm{\parallel }}_X<{\rho }_i$; then both $x,z$ lie within $3{\rho }_i$ of $x_i$, so $\mathrm{\parallel }T(z)-T(x){\mathrm{\parallel }}_Y<\epsilon \mathrm{/}2$. No compact neighborhood of $K$ is asserted or needed.

Apply AP in $X$ to choose bounded finite-rank $P$ with ${\sup }_K\mathrm{\parallel }Px-x{\mathrm{\parallel }}_X<\delta$. By step 1.1, ${\sup }_K\mathrm{\parallel }T(Px)-T(x){\mathrm{\parallel }}_Y<\epsilon \mathrm{/}2$. The set $T(P(K))$ is compact; apply AP in $Y$ to obtain bounded finite-rank $Q$ with ${\sup }_{x\in K}\mathrm{\parallel }QT(Px)-T(Px){\mathrm{\parallel }}_Y<\epsilon \mathrm{/}2$. Thus $QTP$ approximates $T$ on $K$ within $\epsilon$. This makes only two AP selections and does not require a sequence, an infinite union of compact sets, or a choice principle for sequences.

Choose a finite basis ${\alpha }_1,\dots ,{\alpha }_J$ of $P(X)$. Its coordinate functionals are bounded by [F2]; hence $Px={\sum }_j{\mathrm{\ell }}_j(x){\alpha }_j$ with ${\mathrm{\ell }}_j$ bounded on $X$, by composing those functionals with $P$. Write $Rz={\sum }_j{z}_j{\alpha }_j$, $E(x)=({\mathrm{\ell }}_j(x){)}_j$. Choose a finite basis ${\beta }_1,\dots ,{\beta }_s$ of $Q(Y)$, with bounded coordinate functionals $c_i$, and put $h_i(z)=c_i(QT(Rz))$ and $Bv={\sum }_i{v}_i{\beta }_i$. Each $h_i$ is continuous on all ${\mathbb{R}}^J$, and $BhE=QTP$. For a zero-dimensional range use one zero basis vector and the zero coefficient map instead of an actual basis. These formulas prove the stated factorization and complete the proof.