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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Topological_tensor_product#Tensor_products_of_locally_convex_topological_vector_spaces)?



Density and the projective tensor product
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 
$$(p\otimes\_\pi \|\cdot\|)(x\_j\otimes y\_j -d\otimes e) \le p(x\_j-d)\|y\_j\| + p(d)\|y\_j-e\| \le \varepsilon/n.$$


