Examples of Eilenberg-type Swindles I am compiling a list of 'swindles' in the style of the Eilenberg-Mazur swindle. I've already got some swindles in K-theory, the Mazur Swindle and the proof of the Cantor–Bernstein–Schroeder theorem. Can you tell me any more?
 A: The Pelczynski decomposition technique in Banach space theory.
Here's one version:

Let $X$ and $Y$ be Banach spaces such that $X$ is isomorphic to a complemented subspace of $Y$ and $Y$ is isomorphic to a complemented subspace of $X$. Suppose further that $X$ is isomorphic to the $\ell_p$-sum of $X$ with $1 \leq p \lt \infty$:
  $$
X \cong \bigoplus_{n=1}^\infty X = \left\{(x_n) \subset X : \sum_{n=1}^\infty \lVert x_n\rVert^p \lt \infty\right\}.$$
  Then $X$ and $Y$ are isomorphic.

Proof. The hypothesis on $X$ implies that $X \cong X \oplus X$ and the hypotheses on complementation can be written as $X \cong Y \oplus U$ and $Y \cong X \oplus V$. Thus,
$$
Y \cong X \oplus V \cong X \oplus X \oplus V \cong X \oplus Y.
$$
Combining this with the "swindle"
$$
X \cong \bigoplus_{n=1}^\infty X \cong \bigoplus_{n=1}^\infty (Y \oplus U)
\cong  Y \oplus \bigoplus_{n=1}^\infty (U \oplus Y) \cong Y \oplus X,
$$
we get $X \cong Y \oplus X \cong X \oplus Y \cong Y$. $\hskip{1cm}\blacksquare$
This can be used to show that every infinite-dimensional complemented subspace of $\ell_p$ is isomorphic to $\ell_p$: one shows using basic sequence techniques that a closed and infinite-dimensional subspace $Y$ of $\ell_p$ contains a complemented subspace isomorphic to $\ell_p$. The above argument with $X = \ell_p$ then yields the result. This is usually expressed by saying that "$\ell_p$ is prime". 
There are many variants of this argument and lots of applications. See chapter 2 of the book Topics in Banach space theory by Albiac and Kalton for a good exposition. 
