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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?

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    $\begingroup$ Maybe the definition of $\|x\|$ it's just the infimum of $\|x_1\|+\|x_2\|$, without $|\beta_i|$ (otherwise you get $0$, as it is). $\endgroup$
    – Pietro Majer
    Commented Mar 17, 2020 at 21:35
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    $\begingroup$ Oh, I gave that as an exercise in a functional analysis final examination 3 years ago (exercise 2). In case you read Italian... dropbox.com/s/sjo09f70jt9ofpj/soluzioni-IAM-I-2017.pdf?dl=0 $\endgroup$
    – Pietro Majer
    Commented Mar 17, 2020 at 21:41
  • $\begingroup$ Actually, I do read abit of Italian :) $\endgroup$
    – ABIM
    Commented Mar 17, 2020 at 23:13
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    $\begingroup$ This has a simple one line proof using very basic facts about Banach spaces: the map $(x,y) \mapsto x+y$ is a surjection from the product of $X_1$ and $X_2$ onto your space and this displays it as the quotient of a Banach space by a closed subspace $\endgroup$
    – user131781
    Commented Mar 18, 2020 at 6:52
  • $\begingroup$ I know but I needed to save space in a paper (the journal has limits on the paper but not on the appendix). $\endgroup$
    – ABIM
    Commented Mar 18, 2020 at 12:32

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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.

It is the same text.

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  • $\begingroup$ This is fantastic. Thank you very much Andras! :)) $\endgroup$
    – ABIM
    Commented Mar 17, 2020 at 19:09

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