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.



Tensor product space with projective norm is incomplete
The projective tensor product $\ell\_1\widehat{\otimes}X$ is naturally isometrically isomorphic to the $\ell\_1$-sum of countably many copies of $X$. The uncompleted tensor product $\ell\_1 \odot X$ is then the linear span of elements of the form $(\xi\_n x)$, where $(\xi\_n)$ is in $\ell\_1$ under this identification, which is hardly complete as there exist infinite convergent series. For example, take a linearly independent sequence $(x\_n)\_{n=1}^\infty$ of unit vectors in $X$ and consider $(n^{-2}x\_n)\_{n=1}^\infty$; it does not belong to (the image of) $\ell\_1\odot X$.


In general the proof goes along the same lines -- it uses the possibility of (non-unique) representation of elements of the projective tensor product as infinite series of simple tensors. You then have to show that if $X$ and $Y$ are infinite-dimensional then there is an infinite series that cannot be truncated to a finite one.


