Projective tensor product of the normad spaces

Asked Jul 11 '18 at 18:09

Active Jul 11 '18 at 20:23

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How can we show that if $X$ and $Y$ are infinite-dimensional normed spaces, then the projective tensor product of $X$ and $Y$, $X\otimes_\pi Y$, is an incomplete space?

Ps: It is Exercise $2.5$ from Ryan's book (Introduction to tensor products). Unfortunately I did not resolve it.

edited Jul 11 '18 at 20:23

md2perpe

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asked Jul 11 '18 at 18:09

Can you define the projective tensor product? – Federico Fallucca Jul 11 '18 at 18:19
that formula isn't clear how – janmarqz Jul 11 '18 at 19:27
@JanielyMaria. Please use TeX when writing formulas. – md2perpe Jul 11 '18 at 20:10
@FedericoFallucca. I found here that $X \otimes_\pi Y$ is just $X \otimes Y$ endowed with the projective norm $\pi$ defined by $$\pi(u) = \inf { \sum_i |x_i| |y_i| : u= \sum_i x_i \otimes y_i }$$ – md2perpe Jul 11 '18 at 20:22
@JanielyMaria. What have you tried? Do you have any ideas of how it could be done? Do you know what incomplete means? – md2perpe Jul 11 '18 at 20:49

Projective tensor product of the normad spaces

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