Three-space property

Question

I have found two definitions of a three-space property.

One definition is:

$(P)$ is a three-space property if whenever $E$ Banach space, $F\subseteq E$ is a closed linear subspace and two of the space $E$, $F$ and $E/F$ have $(P)$, then all the three spaces have $(P)$.

The other case is:

$(P)$ is a three-space property if whenever $E$ Banach space, $F\subseteq E$ is a closed linear subspace and $F$ and $E/F$ have $(P)$, then $E$ has $(P)$ too.

Are these two definition equivalent?

I have a problem when I try to prove $E/F$ has the property when $E$ and $F$ has the property. Any help would be welcome. Thanks in advanced.

Answer 1

The two definitions are not equivalent.

Let $(P)$ be the property of being infinite-dimensional.

If $F$ and $E/F$ have this property, then $E$ also has this property (it is sufficient that one of $F$ and $E/F$ has that property for $E$ to have it too).

But if $E$ and $F$, resp. $E$ and $E/F$ are infinite-dimensional, then $E/F$ resp. $F$ need not be infinite-dimensional.