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Integrals in tensor product theory

I’m trying to get my head around the tensor product of an infinite dimensional vector space and an infinite dimensional Banach space, with the help of an example I’m reading (sections 3.2.3 and 3.2.4).
In section 3.2.3 I’m told to construct an infinite sequence $x_0,x_1,\ldots$ such that $\|x_n\|=1$, $x_n$ is $n^{th}$ vector of the standard basis for $\ell^2$, $x_n$ is perpendicular to $x_{n-1}$ and finally $x_{ -1}=x_0=e_{1,1}$ (where $e_{i,j}$ is the $i,j$th standard basis vector)
In section 3.2.4 we’re told to check that we can form $\|x_n\otimes y_n\|=1$ whenever $x_n=y_n$.
I’ve constructed this sequence, and computed $\|x_n\otimes y_n\|$:
$\|x_n\otimes y_n\|=\sqrt{\|x_n\|^2\cdot\|y_n\|^2}=\sqrt{n^2\cdot1^2\cdot1^2}=n$
I’ve verified

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