# Density and the projective tensor product

Let $$X$$ be a locally convex space (over $$\mathbb{R}$$), $$D\subset X$$ be dense, $$B$$ be a Banach space (again over $$\mathbb{R}$$) with Schauder basis $$\{b_i\}_{i =1}^{\infty}$$. Is the set $$D^+\triangleq \left\{\sum_{j=1}^n \beta_j d_j\otimes b_j: \, d_j \in D, \, k_j \in \mathbb{R} \right\} ,$$ dense in $$X\otimes E$$ with respect to the projective tensor product?

• Dear user. Interesting point, I made your suggested modification to the question. (btw, I've never heard that expression before, it's very funny :) ) Mar 17, 2020 at 14:19
• Now it’s true. If $E$ and $F$ are lcs’s with dense subsets $E_1$ and $F_1$, then the linear hull of the family of simple tensors they generate is dense in any of the sensible tensor products, in particular for the projective one. By the way, I have deleted my comment, since it is now a nonsense, after the modification. Mar 17, 2020 at 14:43
• True, actually would it still remain true if $b_j$ are taken from a subset of $B$ with dense span instead of from all of $B$? Mar 17, 2020 at 15:02

This seems to be easy, mayby I am missing something: Given dense sets $$D$$ of $$X$$ and $$E$$ of $$B$$, a seminorm $$p$$ on $$X$$, $$\varepsilon >0$$, and $$z=\sum\limits_{j=1}^n x_j \otimes y_j \in X\otimes B$$ it is enough to $$\varepsilon/n$$-approximate with respect to $$p\otimes_\pi \|\cdot\|$$ each term of the sum by an element of $$D \otimes E$$. If $$p(x_j -d)$$ and $$\|y_j-e\|$$ are sufficiently small, we get from the bilinearity $$x_j\otimes y_j -d\otimes e = (x_j-d)\otimes y_j + d \otimes (y_j-e)$$ and hence $$(p\otimes_\pi \|\cdot\|)(x_j\otimes y_j -d\otimes e) \le p(x_j-d)\|y_j\| + p(d)\|y_j-e\| \le \varepsilon/n.$$