I am working with Banach Spaces, which are complete Normed Vector Spaces (NVS).
The norm on a NVS $(E, ||\cdot||_1)$ defines a metric which in turn defines a topology.
Now let us consider $F \subset E$, a set that is itself a NVS with a different norm $||\cdot||_2$.
Can this space $F$ be considered a subspace of $E$? If we suppose that the norm $||\cdot||_2$ cannot be defined in the whole $E$, Does the definition of subspace of a NVS require the norm to be the same?
I need to clarify this concept, because I want to be rigorous. I am facing this situation, in which I have $E \subset F$, $E$ is not closed in $F$ in the topology induced by $||\cdot||_1$, but both are Banach Spaces (i.e both are complete), obviously considering them with their respective norms.