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?
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$\begingroup$ 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. $\endgroup$– user131781Commented Mar 17, 2020 at 14:43
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$\begingroup$ 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 $B$? $\endgroup$– ABIMCommented Mar 17, 2020 at 15:02
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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 $$(p\otimes_\pi \|\cdot\|)(x_j\otimes y_j -d\otimes e) \le p(x_j-d)\|y_j\| + p(d)\|y_j-e\| \le \varepsilon/n.$$
answered Mar 17, 2020 at 14:42