nLab direct sum of Banach spaces

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 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 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 naïve direct sum is not complete under any of these norms, so we must complete it, getting different results for each $p$; this is analogous to the different sequence spaces $l^p$. 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.

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:

$${\Big\| \sum_i a_i i \Big\|} = {\Big\| \sum_i {|a_i|} i \Big\|}$$

(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 $V$-direct sum of this family is a subspace of the direct product of the family, consisting of those $w$ such that this sum converges:

$$\sum_i {\|w_i\|} i$$

(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:

$${\|w\|} \coloneqq {\Big\| \sum_i {\|w_i\|} i \Big\|} .$$

We may succinctly write the $V$-direct sum as follows:

$$\bigoplus^V_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; {\Big\| \sum_i {\|w_i\|} i \Big\|} \lt \infty \Big\} .$$

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

$$\bigoplus^p_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; \sqrt[p] {\sum_i {\|w_i\|^p}} \lt \infty \Big\} ;$$

and if $V = l^\infty$, then

$$\bigoplus^\infty_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; \sup_i {\|w_i\|} \lt \infty \Big\} .$$

These are the $l^p$-direct sum and $l^\infty$-direct sum (which is really a special case). In particular, we have the $l^1$-direct sum:

$$\bigoplus^1_i W_i \coloneqq \Big\{ (w_i)_i \;\Big|\; \sum_i {\|w_i\|} \lt \infty \Big\} .$$

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 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 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 $\mathbb{C}$).

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 integrals. This is particularly common (using $p = 2$) for Hilbert spaces.

References

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

• Kato, Saito, Timura (2003); On $\psi$-direct sums of Banach spaces and convexity; Journal of the Australian Mathematical Society 75, 413–422; web

Here, $\psi$ is the norm, viewed as a convex function of multiple arguments.