Albert Wilansky

On the Proof of the Radon-Nikodym Theorem

The elegant proof (call it P1) of the Radon-Nikodym Theorem, due to von Neumann and given, for example, in [1, Chap. 11, Ex. 37] does not give the result in as much generality as the much less pleasing proof of [1, Chap. 11, Sec. 6]. The purpose of this note is to derive the general result from von Neumanns, thus enabling the elegant proof P1 to be given without loss. Proof P1 yields that if ,i, v are a-finite measures with v << ,i, then there exists f such that v(E) = fEfdpu for all measurable E. This will now be extended so as to drop the assumption that v is a-finite.

A biorthogonal system which is not a Toeplitz basis

We present a Banach space X with a biorthogonal system {bn, f n } , bnÇzX, fn(E.X*f fi(bj) = 8ij, in which {&»} is fundamental, i.e., X is its linear closure, but {bn} is not a Toeplitz basis. We call {bn} a Toeplitz basis if there exists a regular matrix A such that for each x&X, ^2fn(x)bn is ^4-summable to x. I t will be clear that {bn} is not a Toeplitz basis even if the method of summability is considerably more general than a matrix. An example in which X is a non-Banach F space is given in [4, Theorem 13]. Toeplitz bases were introduced in [ l ] , [2], A series ^un is said to be weakly Cauchy if for every continuous linear functional ƒ, Jjf(un) is convergent.