Let $G$ be a compact real Lie group and ${\sf Trig}(G)$ the algebra of trigonometric polynomials on $G$ (defined in the [Hewitt-Ross, Abstract harmonic analysis](http://rads.stackoverflow.com/amzn/click/3540048324), (27.7)), i.e. the algebra of functions $u:G\to {\mathbb C}$ which can be represented as complex linear combinations of functions of the form
$$
x\in G\mapsto \langle U(x)\xi,\eta\rangle\in{\mathbb C},
$$
where $U:G\to B(H)$ is an arbitrary irreducible unitary continuous representation of $G$ (in an arbitrary Hilbert space $H$), and $\xi,\eta\in H$.


Let us call a *tangent vector* on ${\sf Trig}(G)$ in a point $a\in G$ an arbitrary linear functional $f:{\sf Trig}(G)\to{\mathbb C}$ such that:


1) $f$ preserves involution:
$$
f(\overline{u})=\overline{f(u)},\qquad u\in {\sf Trig}(G),
$$ 


2) $f$ satisfies the Leibniz identity in the point $a$:
$$
f(u\cdot v)=u(a)\cdot f(v)+f(u)\cdot v(a), \qquad u,v\in {\sf Trig}(G).
$$


A question: **is it true that every such functional $f:{\sf Trig}(G)\to{\mathbb C}$ is a "true tangent vector" on $G$, i.e. it
can be represented in the form 
$$
f(u)=\lim\_{t\to 0}\frac{u(\gamma(t))-u(\gamma(0))}{t},\qquad u\in {\sf Trig}(G),
$$
for some smooth curve $\gamma:{\mathbb R}\to G$ with $\gamma(0)=a$?**


An equivalent statement: **is it true that every such functional $f:{\sf Trig}(G)\to{\mathbb C}$ can be extended to a functional $h:C^\infty(G)\to{\mathbb C}$ with the same properties?**



Tangent vectors on the algebra of trigonometric polynomials
Yes, of course. The functions you have are representation functions on $G$ (the representations are finite dimensional as well); they are thus algebraic functions on the complexification of $G$. It is well known that the "algebraic Lie algebra is the true tangent space for algebraic groups. 


