Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For a problem I'm working on I have two Banach spaces $X, Y$ and an injective immersion $T\colon X \to Y$ (that is, a $C^1$ injective mapping having the property that its (Fréchet) differential $dT (x)$ is injective at any $x \in X$).

I'm mainly interested in the restriction of $T$ to a finite-dimensional subspace $V_n\subset X$. Can I conclude that $T(V_n)\subset Y$ is contained in a subspace of $Y$ having the same dimension as $V_n$?

Thank you.

share|cite|improve this question
I retagged as functional analysis, as the tag wiki indicates that on Math.SE the (functional-analysis) tag covers both linear and nonlinear aspects. – Willie Wong Nov 21 '12 at 9:04
up vote 1 down vote accepted

Hint:It is false. Please consider the simplest case $X=\mathbb{R}$ and $Y=\mathbb{R}^2$.

share|cite|improve this answer
Oh yes, you could take $T(x)=\gamma(x)$ to be a smooth curve and so disprove the above claim. Thank you. I'll need to broad the question a little to rule out such counterexamples. – Giuseppe Negro Nov 17 '12 at 4:15
@GiuseppeNegro: Yes, please have a try. – 23rd Nov 17 '12 at 4:18
Do you think it could be possible to modify your example to obtain an injective immersion $$ T\colon \mathbb{R}\to \text{some infinite dimensional space}$$ such that its range is not contained in any finite-dimensional subspace? I guess it is, and this would put an end to the question. – Giuseppe Negro Nov 17 '12 at 4:25
@GiuseppeNegro: I think so. – 23rd Nov 17 '12 at 4:41
I definitely was on a false track and you helped me to notice. Thank you very much. – Giuseppe Negro Nov 17 '12 at 16:30

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.