Axiom Of Choice

We have the following indirect implication of form equivalence classes:

40 $\Rightarrow$ 279 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
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
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:

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Implication	Reference
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.

Howard-Rubin Number	Statement
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.