Let's say if we have and we want to show that (open ball of radius ) is convex. We know that (e.g. the norm).
We can still apply the triangle inequality right? So we have: .
In other words, does only apply to ? Or does it apply to general so that we can invoke the triangle inequality? We can use the triangle inequality to prove the general case (which is what I did) right?
I think that you are confusing yourself: the idea of a norm on a linear space with the distance function or metric.
Have a look at these pages:
Norm -- from Wolfram MathWorld
Vector Norm -- from Wolfram MathWorld
In the case of a ‘norm’ as with a metric, the triangle inequality is a requirement.
I think that you need to go to the instructor and clear away this confusion.