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For a locally convex Hausdorff spaces $E$, consider the canonical map $$\overline{\psi}:E^\prime \hat{\otimes}_\pi E \longrightarrow L(E_\sigma)$$ that maps the projective tensor product to the space of weakly continuous maps on $E$ (note, that if $E$ is Banach spaces, then the latter is just the space $L(E)$ of bounded linear operators).

The question whether this map is injective is well-known to be connected to the approximation property. For example:

  • If $E$ is a Banach space, then $E$ has the approximation property if and only if $\overline{\psi}$ is injective.

  • If $E$ has is a complete locally convex space with a fundamental system of absolutely convex neighborhoods $U$ of zero such that every Banach space $E_U$ has the approximation property, then $\overline{\psi}$ is injective.

These statements can be found in Köthe's book "Topologcial Vector Spaces II".

Question: If $E$ is complete and has the approximation property, is it false/true/not known whether $\overline{\psi}$ is injective in general?

Edit: A locally convex space $X$ is said to satisfy the approximation property, if the space of finite rank operators is dense in the space $L_c(X)$, the set of continuous linear operators with the topology of uniform convergence on precompact sets. For Banach spaces, this is equivalent to saying that the finite rank operators are dense in the compact operators with respect to the operator norm.

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  • $\begingroup$ It seems to be $\textit{the approximation propery}$ is not really defined in your post. $\endgroup$
    – Bombyx mori
    Commented Mar 21, 2017 at 3:55
  • $\begingroup$ I added the definition in the post. $\endgroup$
    – Matthias Ludewig
    Commented Mar 21, 2017 at 9:22
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    $\begingroup$ It seems that Grothendieck and also Köthe deal with this characterization of the approximation property ("condition de biunivocite") only in the case of Banach spaces. I suspect that the reason for this is that only in this case you have a "canonical" topology on the dual. In the locally convex case, which topology on $E'$ do you consider to define $E'\hat{\otimes}_\pi E$? $\endgroup$
    – Jochen Wengenroth
    Commented Mar 21, 2017 at 9:35
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    $\begingroup$ Do you have a reference for the equivalence, for a Banach space $X$, between (i) the space of finite rank operators is dense in $L_\sigma(X)$ and (ii) the space of finite rank operators is dense in $K(X)$ for the operator norm? (i)$\implies$(ii) is clear, but I always thought that the converse is not true... $\endgroup$
    – Mikael de la Salle
    Commented Mar 21, 2017 at 9:37
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    $\begingroup$ I do not think that it is a good idea to take the weak$^*$ topology for defining $E'\hat{\otimes}_\pi E$: The completion of $E'_\sigma$ consists of ALL linear functionals on $E$. Perhaps the best choice would be the topology of uniform convergence on all absolutely convex compact sets. $\endgroup$
    – Jochen Wengenroth
    Commented Mar 21, 2017 at 15:47

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