To formulate my question I need the construction of the algebra $J^n_M(K)$ of jets of degree $n$ on a compact set $K$ of a smooth manifold $M$. I'll describe it for the simplest case of $M={\mathbb R}$, $K=[0,1]$ and $n=1$. This is done in two steps.
First, we need the construction of the algebra of polynomials of degree 1 with the coefficients in $C[0,1]$. Let us consider the Banach algebra $C[0,1]$ of continuous functions on $[0,1]$ (with the usual sup-norm), and let us endow its cartesian square $C[0,1]^2$ with the multiplication $$ (x_0,x_1)\cdot(y_0,y_1)=(x_0\cdot y_0,x_0\cdot y_1+x_1\cdot y_0),\qquad x_i,y_i\in C[0,1]. $$ and the norm $$ ||(x_0,x_1)||=||x_0||+||x_1||,\qquad x_i\in C[0,1]. $$ After that $C[0,1]^2$ becomes a Banach algebra. If we introduce an "external variable" $\tau$ with the properties $$ a\cdot \tau=\tau\cdot a, \quad \tau^2=0,\qquad a\in C[0,1], $$ then each element $(x_0,x_1)$ of $C[0,1]^2$ can be represented as a sum $$ (x_0,x_1)=x_0+x_1\cdot\tau, $$ and the multiplication of these sums is exactly the multiplication in $C[0,1]^2$, that is why we call $C[0,1]^2$ the algebra of polynomials of degree 1 with the coefficients in $C[0,1]$.
Let us consider now the algebra $C^\infty(\mathbb R)$ of smooth functions on $\mathbb R$, and the map $$ \varphi:C^\infty(\mathbb R)\to C[0,1]^2\quad\Big|\quad \varphi(x)=(x\big|_{[0,1]},x'\big|_{[0,1]}),\qquad x\in C^\infty[0,1] $$
($x'$ means the usual derivative, and $\cdot\big|_{[0,1]}$ are the restrictions to $[0,1]$). This is a homomorphism of algebras, hence its image $\varphi(C^\infty[0,1])$ is a subalgebra in $C[0,1]^2$. Let us call its closure the algebra of jets of degree 1 of smooth functions on $\mathbb R$ on the subset $[0,1]$ and denote it by $$ J^1_{\mathbb R}[0,1]=\overline{\varphi(C^\infty(\mathbb R))} $$
Being a closed subalgebra in the Banach algebra $C[0,1]^2$, $J^1_{\mathbb R}[0,1]$ itself is a Banach algebra. It is easy to see that $$ J^1_{\mathbb R}[0,1]\ne C[0,1]^2. $$ For example, we can take the pair of functions $$ x_0(t)=t,\quad x_1(t)=0, \qquad t\in [0,1], $$ and we'll see that there is no $x\in C^\infty(\mathbb R)$ such that $$ ||\varphi(x)-(x_0,x_1)||<\frac{1}{4} $$ (if this $x$ exists, then $||x'\big|_{[0,1]}||=||x'\big|_{[0,1]}-x_1||<\frac{1}{4}$, hence $|x(1)-x(0)|=|\int_0^1x'(t)d t|\le||x'\big|_{[0,1]}||<\frac{1}{4}$, but on the other hand, $||x\big|_{[0,1]}-x_0||<\frac{1}{4}$, so $|x(0)|<\frac{1}{4}$ and $|x(1)-1|<\frac{1}{4}$, and as a corollary, $|x(1)-x(0)|>\frac{1}{2}$).
In what I am doing now the following question becomes unexpectedly important:
does $J^1_{\mathbb R}[0,1]$ have the (classical) approximation property?
(In fact, I need the answer for the general case of $J^n_M(K)$, but I think this special question can clarify everything.)
