The Weyl algebra (say, over $\mathbb{C}$) is an universal unital algebra with two generators $x,y$ subject to the relation $xy-yx=1$. This algebra can be constructed in the following way: take two dimensional vector space with basis $\{x,y\}$ and construct the tensor algebra $T(V)$. Then take an ideal $I=(xy-yx-1)$ generated by $xy-yx-1$ and form a quotient $T(V)/I$. It is possible to find a norm on $T(V)$ in which $T(V)$ becomes a normed algebra, therefore we can take the closure $\overline{I}$ of $I$ and form a quotient $T(V)/\overline{I}$. This is again a normed algebra but generators of this algebra satisfy the relation $xy-yx=1$ but this is impossible in any normed unital algebra. This apparent contradiction actually shows that $I$ must be dense in $A$. This leads me to the question: it is really obvious and immediate that $I$ is not the whole $T(V)$? I'm pretty sure that this must be true but it seems to me that it would be hard to give a "two line" proof of this fact. Forgive me if this question is to elementary: but in the literature no one seems to see any problem with the existence (and nontriviality) of Weyl algebra.