The question is as follows (Its not very long - most of it is just definitions!):
A nonnegative real-valued function $\| \cdot \|$ defined on a vector space $X$ is called a pseudonorm if $\|x + y\| \le \|x\| + \|y\|$ and $\|\alpha x\| = |\alpha|\; \|x\|$. Define $x \cong y$, provided $\|x - y\| = 0$. Define $X/\cong$ to be the set of equivalence classes of X under $\cong$ and for $x \in X$ define $[x]$ to be the equivalence class of $x$. Show that $X/\cong$ is a normed vector space if we define $\alpha[x] +\beta[y]$ to be the equivalence class of $\alpha x + \beta y$ and define $\|[x]\|=\|x\|$.
I think there's a bunch of stuff to do here:
Show linearity of classes is well defined
Show classes, so defined, form a vector space
Show the norm is well defined
Show the norm, so defined, satisfies the usual properties
My questions are:
Are these the things to do in the problem?
If so, I can do the last 3, but I'm getting mixed up on the first -- arguing that linearity of classes is well-defined. Usually I start with 2 elements that are in the same class and show the operation defined on one is equal to operation defined on the other. But here we have a binary operation - so I'm confused about how this argument should proceed. It occurred to me that can show $\alpha[x]+\beta[y] = [\alpha x + \beta y]$ as sets but I'm not sure this is sufficient.