TITLE: Tensor product space with projective norm is incomplete
QUESTION [6 upvotes]: Ryan says in his book "Introduction to Tensor Products of Banach Spaces"(pg. 17) that for Banach spaces $X$ and $Y$, $X\otimes Y$ equipped with projective norm is not complete unless $X$ and $Y$ are finite dimensional.
First I want the example of this.
Second, is there any sources about the proof of this statement? Thanks.

REPLY [4 votes]: It wasn't clear to me, immediately, how to do the general case, but I think you can use Biorthogonal Systems.  Indeed, let $(x_n)\subseteq X$ and $(f_n)\subseteq X^*$ satisfy that $f_n(x_m) = \delta_{n,m}$.  Let $(y_n) \subseteq Y$ be a linearly independent sequence.  Set
$$\tau = \sum_{n=1}^\infty \|x_n\|^{-1} \|y_n\|^{-1} 2^{-n} x_n\otimes y_n \in X \widehat\otimes Y. $$
Suppose, towards a contradiction, that $\tau \in X\otimes Y$.  Now, $\tau$ induces a bounded linear map $T:X^*\rightarrow Y$, and by our assumption, $T$ is finite rank.  However, clearly $T(f_n) = \|x_n\|^{-1} \|y_n\|^{-1} 2^{-n} y_n$ and so $(T(f_n))$ is a linearly independent sequence, contradicting $T$ being finite rank.
To construct $(x_n), (f_n)$ we can follow an old argument due to Markushevich.  The argument is not hard, but is a touch long to type out here.  The best link I could find was: Biorthogonal Systems in Banach Spaces (Google books).  Search for "M-basis".