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Let $(X_i)_{i=1}^\infty$ be a sequence of normed spaces. We define the $\ell_p$-direct sum $[\bigoplus_{i=1}^n X_i]_p$ as the normed space of elements $(x_i)_{i=1}^n\in \prod_{i=1}^n X_i$ with norm $$ \|(x_i)_{i=1}^n\|_p=\left(\sum_{i=1}^n\|x_i\|^p\right)^{\frac{1}{p}} $$ for $1\leq p <\infty$ and $$ \|(x_i)_{i=1}^n\|_\infty=\sup_{1\leq i\leq n} \|x_i\|. $$ All these norms are equivalent.

I've been thinking on which properties are inherited by the direct sum from its building spaces (i.e. which properties are such that if they are found in the spaces $X_i$, $i=1,\ldots,n$, then they can be found in $[\bigoplus_{i=1}^n X_i]_p$ too). I've stumbled upon the following (they are properties preserved by isomorphism hence true for any space $[\bigoplus_{i=1}^n X_i]_p$).

  • Completeness (Banach)
  • Separability
  • Reflexivity

We can generalize the definition to consider $[\bigoplus_{i=1}^\infty X_i]_p$ the space of elements $(x_i)_{i=1}^\infty\in \prod_{i=1}^\infty X_i$ such that $$ \|(x_i)_{i=1}^\infty\|_p=\left(\sum_{i=1}^\infty\|x_i\|^p\right)^{\frac{1}{p}}<\infty $$ with norm $\|\cdot\|_p$, $1\leq p< \infty$, and $[\bigoplus_{i=1}^\infty X_i]_\infty$ defined analogously. This is sometimes referred to as the $\ell_p$ sum of spaces $X_i$.

It is known that $[\bigoplus_{i=1}^\infty X_i]_p$ for $1\leq p\leq \infty$ is Banach if the spaces $X_i$ are.
I believe separability is also inherited for $1\leq p <\infty$ since the union of spaces $[\bigoplus_{i=1}^n X_i]_p$, $n\in \mathbb{N}$, is dense in $[\bigoplus_{i=1}^\infty X_i]_p$ (with the trivial embedding).
For the matter of reflexivity, the property is also inherited for $1<p<\infty$ as proved here.

I was wondering if someone could give an aswer regarding this topic more complete than the bits and pieces that I've managed to gather:

Q. Which properties are inherited by the different kinds of direct sums?

Note: If the above properties are true in a direct sum not only if they present in the building spaces but if and only if please tell me so I can edit the question.

Anguepa
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  • Usually, we require all but finitely many components of element in the direct sum to be zero. Do you mean direct product with the $p$-norm? – William Stagner Mar 22 '15 at 03:42
  • We usually do but I am intending to consider the broader case here. – Anguepa Mar 22 '15 at 03:47
  • Then you should replace your notation $\oplus_{i=1}^\infty$ with $\prod_{i=1}^\infty$. – William Stagner Mar 22 '15 at 03:50
  • you should ask which properties you are interested, otherwise question is too broad – Norbert Apr 01 '15 at 13:21
  • Well I would like confirmation that the product of countably many separable spaces is separable. Also, when looking into this topic I lacked a text that addressed it broadly. I look for an answer of this form. – Anguepa Apr 01 '15 at 15:01
  • I'm sorry @WilliamStagner but in the papers I'm reading I've found this notation. I believe it to be not too confusing, given that I state the definition clearly. – Anguepa May 05 '15 at 13:05
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    You might be interested in my answer here. Since $X_i$ is isometrically isomorphic to a closed subspace of the $\ell^p$-sum, completeness, separability and reflexivity are if-and-only-if (with the caveat that the sum can only be separable if at most countably many $X_i$ are nontrivial, and for the reflexivity we need $1 < p < \infty$ or only finitely many nontrivial $X_i$; "only if" holds unconstrained). – Daniel Fischer May 05 '15 at 13:17

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