We have the following indirect implication of form equivalence classes:

202 \(\Rightarrow\) 279
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
202 \(\Rightarrow\) 40 clear
40 \(\Rightarrow\) 43 Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc.
On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 \(\Rightarrow\) 279 All operators on a Hilbert space are bounded, Wright, J.D.M. 1973, Bull. Amer. Math. Soc.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

40:

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

43:

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

279:

The Closed Graph Theorem for operations between Fréchet Spaces: Suppose \(X\) and \(Y\) are Fréchet spaces, \(T:X\to Y\) is linear and \(G=\{(x,Tx): x \in X \}\) is closed in \(X\times Y\). Then \(T\) is continuous. Rudin [1991] p. 51.

Comment:

Back