$\|x+\mathcal{M}\| = \inf\{\|x+y\|: y \in \mathcal{M}\}$ defines a norm Taken From Conway's A course in Functional Analysis chapter 3 Section 4 problem 1:
Problem Statement:
Show that if $\mathcal{M}\leq \mathcal{H}$, then (4.1) defines a norm on $\mathcal{X}/\mathcal{M}$.
I wanted to see if what I had below was complete. Any feedback on rigor is appreciated!
To prove that $\|x+\mathcal{M}\| = \inf\{\|x+y\|: y \in \mathcal{M}\}$ defines a norm, we check that it satisfies the three axioms. We first prove $\|x+\mathcal{M}\| \Rightarrow x \in \mathcal{M}$. We have $\inf\{\|x+y\|: y \in \mathcal{M}\} = 0$, which can only be true if $x\in \mathcal{M}$.
Now we hope to prove that $\|\alpha(x+\mathcal{M})\| =|\alpha|\|(x+\mathcal{M})\|$. We have
\begin{align}
\|\alpha(x + \mathcal M)\| &= \|\alpha x + \mathcal M\| 
\\ & = \inf\{\| \alpha(x+y)\| : y \in \mathcal M\}
\\ & = |\alpha| \inf\{\|x+y\|:y \in \mathcal M\} = 
|\alpha|\cdot \|(x + \mathcal M)\|.
\end{align}
Finally, we prove the triangle inequality, which can be verified with: $\|x+z\| = \inf\{\|x+z+y\|: y \in \mathcal{M}\}= \inf\{\|x+y_1+z + y_2\|: y_1,y_2 \in \mathcal{M}\}\leq \inf\{\|x+y_1\|+\|z + y_2\|: y_1,y_2 \in \mathcal{M}\} =  \inf\{\|x+y_1\|: y_1 \in \mathcal{M}\} +  \inf\{\|z + y_2\|:y_2 \in \mathcal{M}\}$.
 A: I'm not sure about notation, but based on the problem context I'd wager that $\mathcal{M}\leq\mathcal{H}$ might be supposed to mean $\mathcal{M}$ is a closed vector subspace of $\mathcal{H}$? Assuming this is the case, here are some hints/comments on the 3 parts of your proof.

*

*I think you're missing an "$=0$" in the $\|x+\mathcal{M}\|\Rightarrow x\in \mathcal{M}$ part. Also, the argument you've stated is a tautology. Hint: What does it actually mean for the infimum to be zero? Also, is $\mathcal{M}$ closed? If so, this could be quite useful in proving this part.


*The positively homogeneous part (involving $\alpha$) is only true if $\mathcal{M}$ is positively homogeneous. For instance, if $\mathcal{M}$ is a vector subspace then this would be satisfied, but you may want to state the reasoning.


*The triangle inequality part looks OK to me, however I would also note that you may want to write a justification (in words) as to why you can express an arbitrary $y\in\mathcal{M}$ as $y_1+y_2$ where $y_1,y_2\in\mathcal{M}$. Again, if $\mathcal{M}$ is a vector subspace then this is true. However, we have to be careful, because this is not true for arbitrary sets.
