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For Banach spaces $X$ and $Y$, we call $\pi$ the projective norm on $X \otimes Y$ which is defined as

$$ \pi (u) = \inf \left\{ \sum_{i=1}^{n} \| x_i \| \| y_i \|: u = \sum_{i=1}^{n} x_i \otimes y_i \right\}. $$

The textbook (Raymond. A. Ryan) explains the choice of the name "projective" by introducing a quotient operator.

$Q : Z \rightarrow Y$ is called a quotient operator if $Q$ is surjective and $\| y \| = \inf \{ \| z\| : z \in Z, \, Qz = y \}$ for every $y \in Y$. This is equivalent to that $Q$ maps the open unit ball of $Z$ onto the open unit ball of $Y$.

Thus, if $Q : Z \rightarrow Y$ is a quotient map, then $Y$ is isometrically isomorphic to the quotient space $Z/\ker Q$.

I have checked the above facts about quotient operators, but I couldn't see the origin of the term.

What is the reason that we use the term "projective"?

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The relevant quotient appears in the definition of tensor product. We begin with "the free vector space" $F$ which is the space of formal linear combinations

$$ \sum_{ij} c_{ij} x_i y_j$$

with coefficients $c_{ij} $ in the scalar field, $x_i\in X$, and $y_j\in Y$. The basis of this giant space consists of all $(x_i, y_j)$ pairs.

The space $F$ is naturally normed as

$$ \left\| \sum_{ij} c_{ij} x_i y_j \right\| = \sum_{ij} |c_{ij}| \|x_i\| \|y_j\| $$

Then we build tensor space $X\otimes Y$ as a certain quotient of $F$, identifying elements $c(xy) \sim (cx)y \sim x(cy)$, as well as $xy + zy \sim (x+z)y$ and $xy+xz \sim x(y+z)$.

The quotient norm on $X\otimes Y$ is the projective norm $\pi$ defined above: it's just the infimum of norms of all free-space elements that project to a given element of $X\otimes Y$.

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  • $\begingroup$ If I understand your comment correctly, the term "projection" came from the projection from the free vector space $F$ (of pair $(X,Y)$) to $X \otimes Y$. Right? $\endgroup$ – cdamle Dec 8 '17 at 1:42
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    $\begingroup$ Yes, it's from that projection (i.e., quotient map) $\endgroup$ – user357151 Dec 8 '17 at 2:06

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