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For the proof, the existence of a linear surjective norm preserving mapping (L : {b_{c}^{r,s} }^longrightarrowell_{1}) should be shown.

Given any two almost orthonormal systems A and ˜A of unit vectors in a Hilbert space X of the same cardinality, there exists a norm preserving almost isometry F of X so that F(A)="A˜.

Therefore, L is norm preserving.

So, L is norm preserving.

Hence, T is surjective and norm preserving.

Consequently, T is surjective and is norm preserving.

end{aligned} That is, T is norm preserving.

Since begin{aligned} ||Bvarphi ||^2=langle varangle^2varphi rangle =langle varphi, 1-A^2)varphi rangle = 1-lambda ^2)||varphi ||^2 end{aligned}we see that V is norm preserving.

end{aligned} Hence, we conclude that L is norm preserving and (x in b^{r,s}_{p}), namely L is surjective.

Moreover, one can easily see for every x ∈ c 0 λ ( B ˜ ) that ∥ T x ∥ ∞ = ∥ y ∥ ∞ = ∥ Λ ˜ x ∥ ∞ = ∥ x ∥ c 0 λ ( B ˜ ), which means that T is norm preserving.

Proof of Theorem 1.4 Since f : E 1 → E 2 is a random isometry with f ( 0 ) = 0, we see that f is random norm preserving and f | L 2 ( E 1 ) is a mapping from L 2 ( E 1 ) to L 2 ( E 2 ).