# Definitions of projective tensor norm

The Proposition 2.8 of Ryan's textbook(Introduction to tensor products of Banach spaces) says: Let $$X$$ and $$Y$$ be a Banach space. Let $$u \in X\hat\otimes_{\pi} Y$$ and $$\epsilon >0$$. Then there exist bounded sequences $$(x_n), (y_n)$$ in $$X, Y$$ respectively such that the series $$\sum_{n=1}^{\infty} x_n \otimes y_n$$ converges to $$u$$ and $$\sum_{n=1}^{\infty} \|x_n\|\|y_n\|<\pi(u)+\epsilon.$$

From this we have that the projective norm $$\pi (u)$$ is that $$\pi(u)=\inf\left\{\sum_{n=1}^{\infty} \|x_n\|\|y_n\|:\sum_{n=1}^{\infty} \|x_n\|\|y_n\|<\infty,\, u = \sum_{n=1}^{\infty} x_n \otimes y_n \right\},$$ the infimum being taken over all the representations of $$u$$.

Question: Could you explain why $$\pi(u) = \inf\left\{\sum_{n=1}^{\infty} |\lambda_n| \|x_n\|\|y_n\|: u = \sum_{n=1}^{\infty} \lambda_n x_n \otimes y_n, \, \sum_{n=1}^{\infty} |\lambda_n|<\infty,\, x_n, y_n \rightarrow 0\right\}?$$

I have tried to show this by multiplying $$x_n, y_n$$ by some values to find the desired representation of $$u$$ (which is of the form as in the statement in Question), but I couldn't get it.

• What is $X\hat\otimes_{\pi} Y$? – Paul Sinclair Nov 15 '19 at 2:30
• @PaulSinclair It is the completion of the normed space $X\otimes Y$. – cdamle Nov 15 '19 at 6:08

Try showing that if $$u = \sum_{n=1}^{\infty} x_n \otimes y_n$$ and $$\sum_{n=1}^{\infty} |\lambda_n|<\infty$$ for some sequence with $$\lambda_n \ne 0$$. Then if you let $$\hat x_n = (1/\lambda_n)x_n$$, it must be true that both $$\hat x_n \to 0$$ and $$y_n = 0$$.
• If $\hat{x_n}=(1/\lambda_n) x_n$, how could you see that $\hat{x_n} \rightarrow 0$? – cdamle Nov 15 '19 at 6:10