 Pełczyński decomposition method

ID: pelczynski-decomposition-method

E

satisfying

E ≅ (⨁_{n \geq 1} E)_{p}

W

is absorbed by

E

: write

E ≅ W \oplus V

, expand the countable sum, and absorb one additional copy of

W

into the countably many

W

summands. Hence

E \oplus W ≅ E

. If

Z

is complemented in

E

and contains a complemented copy of

E

, write

Z ≅ E \oplus W

with

W

complemented in

E

; the preceding argument gives

Z ≅ E

. The decompositions are Banach space isomorphisms, not merely algebraic decompositions.

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