Description:
In this note we give the definitions for forms
Form 279 and Form 281 and related results from
Wright [1973].
Content:
In Wright [1973] it is shown that Form 0\(\not\Rightarrow\)Form 281 by showing that every linear operator on a Hilbert space is bounded in Solovay's model
\(\cal M 5\). In this note we give the definitions for forms
Form 279 and Form 281 and related results from
Wright [1973].
Definition: Suppose \(X\) is a topological vector space.
- \(X\) is locally convex if there is a neighborhood base at \(\vec{0}\) consisting of convex sets. (A subset \(S\) of \(X\) is convex if for all \(x\) and \(y\) in \(S\), \(L(x,y)\subseteq S\), where \(L(x,y)= \{\lambda_1 x + \lambda_2 y:\lambda_1, \lambda_2\in \Bbb R, \lambda_1 + \lambda_2 = 1,\hbox{ and, } 0\le \lambda_1,\lambda_2\le 1\}\).
- A metric \(d\) on \(X\) is invariant if for all \(x,\ y\) and \(z\) in \(X\), \(d(x+z,y+z) = d(x,y)\).
- \(X\) is a Fréchet space if \(X\) is locally convex and the topology on \(X\) is induced by a complete, invariant metric.
In the model \(\cal M 5\) (Solovay [1970]) Form 43 \((DC)\) and the negation of Form 280 hold (the negation of Form 280 is "Every subset of a complete, separable metric space has the Baire property."). Wright [1973] shows that the following are consistent by showing that they are theorems of \(ZF + DC + \neg\)Form 280:
- If \(X\) and \(Y\) are separable, metrizable topological groups and \(X\) is complete and \(H: X \to Y\) is a group homomorphism, then \(H\) is continuous.
- If \(X\) is a Fréchet space and \(Y\) is a separable, metrizable topological vector space and \(T: X \to Y\) is linear, then \(T\) is continuous.
- Every linear functional on a Fréchet space is continuous.
- If \(X\) and \(Y\) are Fréchet spaces, \(T: X \to Y\) is linear and there are enough linear functionals on \(Y\) to separate the points of \(Y\) then \(T\) is continuous.
- If \(H\) is a Hilbert space and \(T : H \to H\) is linear, then \(T\) is bounded.
Howard-Rubin number:
96
Type:
Definitions and summaries
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