Rudin's statement of Hahn-Banach's theorem

Question

Rudin's statement of the Hahn-Banach theorem in Functional Analysis (Theorem 3.2) involves a linear form $f$ defined on some subspace of a real vector space, and which is bounded by a sublinear function, as usual. But the specific hypotheses assumed about this sublinear function are:

$p(x+y) \leq p(x)+p(y)$

For non-negative $t$, $p(tx)=p(x)$ (!!)

Shouldn't it be $p(tx)=tp(x)$? The proof seems to implicitly use the latter condition, and not condition (2) quoted above. But I can't find references to any errata online.

Is the theorem still true with this different assumption on $p$? If so, how do you deduce from it that it also works when $p$ is a semi-norm? Rudin basically just says that if $p$ is a semi-norm, the result is "contained in Theorem 3.2", but a semi norm does not satisfy $p(tx)=p(x)$.

Question of weakening hypothesis, which is always rewarding in mathematics? — Piquito, Sep 29 '16 at 15:04
The assumption that $p(tx)=p(x)$ for all $t \ge 0$ makes it easier because $p(x)=p(0)$ for all $x$. Definitely simplifies everything, but it's not very useful. — Disintegrating By Parts, Sep 29 '16 at 15:50

score 7 · Accepted Answer · answered Oct 24 '16 at 22:56

7

Yes, I believe it is a typo. I had a look at a French translation of the book today, and it has $p(tx)=tp(x)$, as expected.

answered Oct 24 '16 at 22:56

Jack M

27,819
7
63
129

Rudin's statement of Hahn-Banach's theorem

1 Answers1