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\section*{direct sum of Banach spaces}

\hypertarget{direct_sums_of_banach_spaces}{}\section*{{Direct sums of Banach spaces}}\label{direct_sums_of_banach_spaces}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{direct_integrals}{Direct integrals}\dotfill \pageref*{direct_integrals} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of [[direct sum]] extends easily from [[vector spaces]] to [[topological vector spaces]]; we wish to explore a similar but more general notion in the case of [[Banach spaces]].

When taking the direct sum of two (or any [[finite set|finite number]]) of Banach spaces (i.e., in the category of Banach spaces and continuous linear maps), the only question is which [[norm]] to use; and we have a choice, entirely analogous to the choice of norms to put on the [[Cartesian space]] $\mathbb{R}^2$ (and its [[complexification|complexified]] variant $\mathbb{C}^2$): one for each [[extended real number]] $p \in [1, \infty]$ (and actually more choices than that). In fact, this is a special case, the direct sum of two copies of the [[line]] $\mathbb{R}$ or $\mathbb{C}$.

For infinitely many summands, the na\"i{}ve direct sum is not [[complete space|complete]] under any of these norms, so we must complete it, getting different results for each $p$; this is analogous to the different [[sequence space]]s $l^p$. Again, this is a special case, a direct sum of infinitely many copies of the line.

In accordance with the last analogy, we speak of $l^p$-direct sums. In fact, even more variety is possible, corresponding to other possible norms on standard Banach spaces.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $V$ be a [[Banach space]] equipped with a [[Schauder basis]] $B$, so that every element of $V$ may be written uniquely as an infinitary [[linear combination]] of elements of $B$. Suppose also that that the basis is normal: the norm of any element of $B$ is $1$; and absolute:

\begin{displaymath}
{\Big\| \sum_i a_i i \Big\|} = {\Big\| \sum_i {|a_i|} i \Big\|}
\end{displaymath}
(where the $i$ are the elements of the basis and the $a_i$ are scalars). The typical example is the [[sequence space]] $l^p$ (or a finitary or uncountablary version) with its usual basis.

Then given a [[family]] $W$ of Banach spaces indexed by the [[set]] $B$, the \textbf{$V$-direct sum} of this family is a [[subspace]] of the [[direct product]] of the family, consisting of those $w$ such that this sum converges:

\begin{displaymath}
\sum_i {\|w_i\|} i
\end{displaymath}
(where again the $i$ are the basis vectors and $w_i$ is in the space $W_i$, with the norm of $w_i$ taken in $W_i$ and the sum taken in $V$). Then the norm of $w$ is the norm of this sum:

\begin{displaymath}
{\|w\|} \coloneqq {\Big\| \sum_i {\|w_i\|} i \Big\|} .
\end{displaymath}
We may succinctly write the $V$-direct sum as follows:

\begin{displaymath}
\bigoplus^V_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; {\Big\| \sum_i {\|w_i\|} i \Big\|} \lt \infty \Big\} .
\end{displaymath}
Strictly speaking, the only condition on the right-hand side is that the sum exists in $V$; then of course its norm will be finite. However, often some sense can be established for the sum outside of $V$ but then it will have no (finite) norm.

In particular, if $V = l^p$ (or a finitary or uncountablary version of such) for $1 \leq p \lt \infty$, then

\begin{displaymath}
\bigoplus^p_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; \sqrt[p] {\sum_i {\|w_i\|^p}} \lt \infty \Big\} ;
\end{displaymath}
and if $V = l^\infty$, then

\begin{displaymath}
\bigoplus^\infty_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; \sup_i {\|w_i\|} \lt \infty \Big\} .
\end{displaymath}
These are the \textbf{$l^p$-direct sum} and \textbf{$l^\infty$-direct sum} (which is really a special case). In particular, we have the \textbf{$l^1$-direct sum}:

\begin{displaymath}
\bigoplus^1_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; \sum_i {\|w_i\|} \lt \infty \Big\} .
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

If [[short linear maps]] are taken as the [[morphisms]] in the category of Banach spaces, then the $l^1$-direct sum is the [[coproduct]], and the $l^\infty$-direct sum is the [[product]].

We can also consider the abstract concepts of [[direct sum]] and [[weak direct product]]; here again the $l^1$-direct sum is the direct sum, and the $l^\infty$-direct sum is the weak direct product. (It is quite common for coproduct and direct sum to be the same, but weak direct product usually diverges from the product for infinitely many objects. That they match up here crucially depends on [[complete space|completeness]].)

If every Banach space in a direct sum is a [[Hilbert space]], then their $l^2$-direct sum is also a Hilbert space. This is the standard notion of \textbf{direct sum of Hilbert spaces}. In [[Hilb]], this the abstract [[direct sum]], the [[weak direct product]], and the [[coproduct]]. Thus for finitely many objects, it is a [[biproduct]] (so $Hilb$ behaves rather like [[Vect]]).

Any Banach space $V$ with basis $B$ is the $V$-direct sum of ${|B|}$ copies of the line ($\mathbb{R}$ or $\mathbb{C}$).

\hypertarget{direct_integrals}{}\subsection*{{Direct integrals}}\label{direct_integrals}

As $l^p$ is the [[Lebesgue space]] $L^p$ for a [[measure space]] with [[counting measure]], and infinitary sums are simply the [[integrals]] on such a measure space, we may generalise from direct sums of Banach spaces to their [[direct integral]]s. This is particularly common (using $p = 2$) for [[Hilbert spaces]].

\hypertarget{references}{}\subsection*{{References}}\label{references}

Here's something about direct sums of finitely many Banach spaces using norms (on $\mathbb{C}^n$) \emph{other} than the usual $l^p$-norms:

\begin{itemize}%
\item Kato, Saito, Timura (2003); On $\psi$-direct sums of Banach spaces and convexity; Journal of the Australian Mathematical Society 75, 413--422; \href{http://www.austms.org.au/Publ/Jamsa/V75P3/n57.html}{web}

\end{itemize}
Here, $\psi$ is the norm, viewed as a [[convex function]] of multiple arguments.

[[!redirects direct sum of a Banach space]] [[!redirects direct sums of a Banach space]] [[!redirects direct sum of Banach spaces]] [[!redirects direct sums of Banach spaces]]

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