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Let $Q:X\to Y$ be a quotient map, i.e. a map between normed spaces such that $B_Y(0,1)=Q(B_X(0,1))$ ($Q$ maps the unit balls onto each other). Then $Q$ is surjective and $Q\in L(X,Y)$ with $\|Q\|=1$.

I already showed that $Q$ is surjective:

Let $y\in Y$, then define $\tilde y=\frac{y}{\|y\|}$ and there exists $\tilde x\in B_X(0,1)$ such that $\tilde y=Q(\tilde x)$. Using the linearity we get $$y=\|y\| \tilde y =\|y\|Q(\tilde x)=Q(\|y\|\tilde x)=Q(x) $$ with $x:=\|y\|\tilde x$ and hence $Q$ is surjective.

Also we have $\sup_{x\in B_X(0,1)}\|Qx\|_y=\sup_{y\in B_Y(0,1)}\|y\|_Y \geq 1$ since $S^1_Y = \{y\in Y: \ \|y\|=1\}\subset B_Y(0,1)$, and $\|Qx\|_Y\leq 1$ since $Qx\in B_Y(0,1)$ and hence $\|Q\|=1$.

Now I only have to prove that $Q$ is linear, but I have no idea how to do this. Can anyone help me out? Thanks!

dinosaur
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  • The linearity of $Q$ must be part of the hypothesis. Without that, from $B_Y(0,1) = Q(B_X(0,1))$, nothing further can be deduced. – Daniel Fischer May 27 '14 at 14:48

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This is an illustration of how using symbols instead of words leads to misunderstanding. By asking you to show $Q\in L(X,Y)$ with $\|Q\|=1$ the author meant that you should check that $Q$ is bounded (and, specifically, has norm $1$). The linearity was implicitly understood from the beginning:

a quotient map, i.e. a linear map between normed spaces such that

The proof you gave is correct.

  • We did not define quotient maps as linear maps, just as maps, which map the unit ball in $X$ onto the unit ball in $Y$, but all the books and websites I checked for research on how to prove linearity say, quotient maps are "linear maps such that ...". So I assume the definition we got in class is not correct? – dinosaur May 27 '14 at 19:37
  • @dinosaur Linearity should be assumed; if it was not, that's an omission. Otherwise the map $f:\mathbb R^2 \to \mathbb R$ defined by $$f(x,y) = \begin{cases} x ,\quad &|x|\le 1 \ y^{42}\sin x , \quad & |x|>1 \end{cases}$$ would qualify as a projection. –  May 27 '14 at 19:40
  • okay, thank you! – dinosaur May 27 '14 at 19:41