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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.

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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
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up vote 1 down vote accepted

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

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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
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@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
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