I am interested in solving Exercise 2.4 from this set of notes I found online. The exercise reads:
Exercise 2.4: Let $M$ be a closed subspace of a Hilbert space, $H$. Define a map $T:M^\perp\to H/M$ by $T(x)=x+M$. Prove that $T$ is an isometric isomorphism of $M^\perp$ onto $H/M$.
I have included my attempt below, and am looking for help regarding the parts I have not been able to complete. I would also appreciate feedback on tidying up my attempt so far – is there anything which can be better explicated? Or anything that I have missed out? I am quite new to dealing with quotients of Hilbert spaces and things like cosets; is there a standard reference in which quotient space results like this are proven in full?
My attempt: I need to show that (1) $T$ is linear, (2) that $T$ is an isometry and (3) that $T$ is onto. I understand that if $T$ is linear, then additionally being an isometry yields injectivity. Hence, we then only need to show that $T$ is surjective. I show (2) last, since this is the part I am currently stuck on.
(1) Let $f,g\in M^\perp$. Then
$$T(f+g)=(f+g)+M=(f+m)+(g+M)=T(f)+T(g),$$
by the definition of addition of cosets in $H/M$. Thus, $T$ is linear.
(3) By the Hilbert space decomposition of $H=M\oplus M^\perp$, for any $x\in H$ there exists a unique $u_x\in M$ and $v_x\in M^\perp$ such that $x=u_x+v_x$. With this in mind, let $x+M\in H/M$ with $x\in H$ arbitrary. Since the Hilbert space decomposition is unique, we can identify this $x$ by it's corresponding $v_x\in M^\perp$ part in the decomposition. This proves the surjectivity of $T$, since we can map to any $x+M\in H/M$ by $v_x\in M^\perp$.
(2) Let $f\in M^\perp$. Then
$$\|T(f)\|_{H/M}=\|f+M\|_{H/M}=\inf_{m\in M}\|f-m\|_H\le\|f-0\|_H\le\|f\|_H,$$
since $M$ is a linear subspace and contains in particular $m=0$. As for the other direction, I am not so sure. I found the question $Q:\mathbb{M}^{\perp}\rightarrow \mathbb{H}/\mathbb{M}$ is an isometric isomorphism and noticed the initial comment given there. I understand how Pythagoras' Theorem has been used together with the Hilbert space decomposition I used in (3) above. But what I don't understand is how, exactly, it is used with the norm on the quotient space. Being unfamiliar with cosets and quotient spaces, I am also not entirely sure how to translate it to the form of the norm I have here.
Could somebody help me to bridge the gap here, please? How do I translate from the ideas sketched there, to the scenario as I have phrased it here?
Best Answer
If $x \in M$ and $y \in M^{\perp}$ then $\|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}$. Hence, $\|x+M\|^{2}=\inf \{\|x+y\|^{2}: y \in M\} \geq \|x\|^{2}$. This proves that $\|Tx|| \geq \|x\|$.
[I have just used the definition of the norm in $X/M$ and the fact that if each eelment of a set of real numbers is greater than or equal to $c$ then the infimum of the set is greater than or equal to $c$].