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"?


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 $X\otimes Y$.

  • $\begingroup$ 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
  • 1
    $\begingroup$ Yes, it's from that projection (i.e., quotient map) $\endgroup$ – user357151 Dec 8 '17 at 2:06

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