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53.22 53. Appendix: Projective resolutions of identity on Banach spaces 595

53.22. Theorem. [Bartle, Graves, 1952] Let T : E ’ F be a bounded linear
surjective mapping between Banach spaces. Then there exists a continuous mapping
S : F ’ E with T —¦ S = Id.

Proof. By the open mapping theorem there is a constant M0 > 0 such that for all
y ¤ 1 there exists an x ∈ T ’1 (y) with y ¤ M0 . In fact there is an M0 such
that B1/M0 ⊆ T (B1 ) or equivalently B1 ⊆ T (BM0 ). Let (fγ )γ∈“ be a continuous
partition of unity on oF := {y ∈ F : y ¤ 1} with diam(supp(fγ )) ¤ 1/2. Choose
xγ ∈ T ’1 (carr(fγ )) with xγ ¤ M0 and for y ¤ 1 set

S0 y := fγ (y)xγ and recursively
Sn (an (y ’ T Sn y)),
Sn+1 y := Sn y +
where an := 22 .
By induction we show that the continuous mappings Sn : {y : y ¤ 1} ’ E satisfy
y ’ T Sn y ¤ 1/an and Sn y ¤ Mn := M0 · k=0 (1 + 1/ak ).
(n = 0) Obviously S0 y ¤ fγ (y) xγ ¤ M0 and

y ’ T S0 y = fγ (y)(y ’ T xγ ) ¤ fγ (y) y ’ T xγ ¤ = a0 ,
γ γ∈“y

where “y := {γ ∈ “ : fγ (y) = 0}.
(n + 1) For y ¤ 1 and yn := an (y ’ T Sn y) we have yn ¤ 1 by induction
hypothesis. Then

1 1
Sn+1 y ¤ Sn y + S n y n ¤ Mn + Mn = Mn+1 .
an an

y ’ T Sn+1 y = y ’ T Sn y ’ T Sn (an (y ’ T Sn y))
1 1 1 1
¤ y n ’ T S n yn ¤ · = .
an an an an+1

Now (Sn ) is Cauchy with respect to uniform convergence on {y : y ¤ 1}. In fact

1 Mn M∞
Sn+1 y ’ Sn y ¤ Sn (an (y ’ T Sn y)) ¤ ¤ ,
an an an

where M∞ := limn Mn . Thus S := limn Sn is continuous and y’T Sy = limn y’
T Sn y = 0, i.e. T Sy = y. Now S : F ’ E de¬ned by S(y) := y S( y ) and
S(0) := 0 is the claimed continuous section.



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