Let $X\_1,X\_2$ be Banach subspaces of a locally convex space $X$. Then the subset
$$
X\_1+X\_2 = \left\{
x\in X:\, x= \beta\_1 x\_1 + \beta\_2 x\_2 \, \beta\_i \in \mathbb{R},\, x\_i \in X\_i
\right\},
$$
a is a normed space with respect to the norm
$$
\|x\| := \inf\left\{
\sum\_{i=1}^2|\beta\_i|\|x\_i\|\_i: x = x\_1 +x\_2, \beta\_i \in \mathbb{R}
\right\}
$$
where $\|x\_i\|\_i$ is the norm of the element $x\_i \in X\_i$ and the infimum is taken over all representations of $x$ as the sum of elements in $X\_1$ and $X\_2$. Furthermore, the closure of $X\_1+X\_2$ in $X$ is Banach with respect to the completion of the norm $\|\cdot\|$. 


What's a reference to this fact?



Reference request: completion of Banach norm on sum
You find it in the manuscript of Alessandra Lunardi on Interpolation Theory. A free version is available here: <http://people.dmi.unipr.it/alessandra.lunardi/LectureNotes/SNS1999.pdf>


In these notes the argument is (somewhat compressed) on page 3.


If you want a ZBL reference, then see page ix in


*Lunardi, Alessandra*, Interpolation theory, Appunti. Pisa: Scuola Normale Superiore. 149 p. (1999). [ZBL1165.41300](https://zbmath.org/?q=an:1165.41300).


It is the same text.


