What Does "Complemented Copy" of a Space Mean?

Question

What does mean a "complemented copy" of a space? I saw this term in many papers, but could not find the definition. For instance:

A Banach space $X$ contains a complemented copy of $\ell_1$.

Many thanks.

Answer 1

A (closed) subspace $Y$ of a linear space $X$ is called complemented when there is a (closed) subspace $Y'$ of $X$ such that $Y' \cap Y= \{0\}$ and $Y' + Y=X$ and the projection $P: X \to Y$ sending $x$ to $y_1 \in Y$ (where $X=y_1+y_2, y_1 \in Y, y_2 \in Y'$ in a unique way) is bounded. See mathworld for a full definition and basic facts. It's a generalisation of orthogonal complement (from inner product spaces and Hilbert spaces) to a more general setting.

$X$ has a complemented copy of $\ell_1$ when such a $Y$ exists that is isometric (or sometimes also just isomorphic is enough) to $\ell_1$.

Answer 2

Definition 4.20 from Rudin's Functional Analysis might be relevant:

Suppose $M$ is a closed subspace of a topological vector space $X$. If there exists a closed subspace of $N$ of $X$ such that $X = M+N$ and $M \cap N = \{0\}$, then $M$ is said to be complemented in $X$. In this case, $X$ is said to be the direct sum of $M$ and $N$, and the notation $X = M \oplus N$ is sometimes used.

This is just a guess, but I suppose that $X$ containing a complemented copy of $\ell_1$ means that it contains an isomorphic copy of $\ell_1$ (call it $M$) as a closed subspace and that there exists a subspace $N$ satisfying the above definition such that $X = M \oplus N$ (or $X = \ell_1 \oplus N$, if you can tolerate the abuse of notation).