Tietze extension theorem

Stream: maths

[Yury G. Kudryashov]

I'm formalizing Tietze extension theorem. Most sources state if for f : s → ℝ. Clearly, it is true for f : s → (ι → ℝ) and f : s → E, where E is a finite dimensional normed space over ℝ. Are there more general versions? E.g., is it true for f : s → E, where E is a Banach space?

[Scott Morrison]

https://mathoverflow.net/questions/201704/between-tietzes-and-dugundjis-extension-theorems

[Scott Morrison]

Oh, maybe that's not quite it. I thought Bill Johnson's answer was addressing your question, but now I think the hypotheses may be different.

[Yury G. Kudryashov]

BTW, it seems that the proof at ncatlab in the case of an unbounded function has a mistake (or I miss something trivial).

Finally consider the case that ff is not a bounded function. In this case consider any homeomorphism $\phi \colon \mathbb{R}^1 \overset{\simeq}{\to} (-c_0,c_0) \subset \mathbb{R}^1$ between the real line and an open interval. Then $\phi \circ f$ is a continous function bounded by $c_0$ and hence the above argument gives an extension $\widehat {\phi \circ f}$. Then $\phi^{-1} \circ \widehat{ \phi \circ f }$ is an extension of $f$.

Why $\widehat {\phi \circ f}$ can't take value $c_0$ at some point?

[Scott Morrison]

https://mathoverflow.net/questions/136554/generalizations-of-the-tietze-extension-theorem-and-lusins-theorem?rq=1 seems to be about your question.