Let $X$ and $Y$ be Banach spaces (algebras). If $X\otimes^\pi Y$ denotes the projective tensor product of $X$and $Y$, define $P:X\otimes^\pi Y\rightarrow X$ as follows : for $x\otimes y\in X\otimes Y$, let $P(x\otimes y):= x$. Is true that $P$ is a projection? and how can I prove that.
I need to prove that if $(t_\alpha)$ is a bounded net in $X\otimes^\pi Y$ (where for each $\alpha$ , $t_\alpha:= \sum_{i=1}^\infty \ x_i\otimes y_i$ for some $x_i\in X$ and $ y_i\in Y$ for all $i$), then the sequence $(x_i)$ is bounded in $X$.
Thank you in advance
$P$ isn't even well-defined. For the corresponding map $\tilde P \colon X \times Y \to X$, $\tilde P(x,y) = x$ is not bilinear (if $X \ne 0$). If $\tilde P$ is bilinear and hence $P$ well-defined, we must have
$$ x = P\bigl(x \otimes (y_1+y_2)\bigr) = P(x \otimes y_1 + x \otimes y_2) = P(x\otimes y_1) +P(x \otimes y_2) = 2x $$ for all $x$, hence $X = 0$.
So $P$ is a well-defined projection iff $X=0$, but I'm sure, that is not the case you are interested in.