Polynomial

[Camillo]

Prove that every polynomial is a tempered distribution.

[elgiovo]

"Camillo":[quote="elgiovo"]

Your solution is not clear to me .

Can you provide more details and explanations ?
Thanks[/quote]

Sure. Here is some more insight.
First of all, some notions about the topology of $ccS(RR)$: it is the vector space of $C^(oo)$ functions $phi$ on $RR$ s.t. $x^(alpha)D^(beta) phi(x)$ is bounded $forall(alpha, beta) in NN_0^2$ (here $D=1/i ("d")/("d"x)$).
$ccS(RR)$ is provided with the family of seminorms

$p_M(phi)="sup"{^M|D^(alpha)phi(x)|; x in RR, alpha
which turn $ccS(RR)$ into a Fréchet space.

Now, $ccS'$ consists of the linear functionals $Lambda$ on $ccS$ for which there exists an $M in NN_0$ and a constant $C_M$ (depending on $Lambda$) such that

$|Lambda(phi)|<=C_M * p_M(phi)$, $forall phi in ccS$.

This is the consequence of an important theorem in the theory of Fréchet spaces.
Now you've got all the ingredients to understand the proof: in fact what I had shown is

$|p(phi)|=||<=C' cdot p_(N+2)(phi)$, $forall phi in ccS$.

[Camillo]

"elgiovo":

Your solution is not clear to me .

Can you provide more details and explanations ?
Thanks

[Camillo]

Here another proof

If $u $ is a polynomial and $v in ccS$, after choosing a polynomial $P$ such that $u/P in ccL^1 $, we have :

$ |int_(RR^n) uv| =|int_(RR^n) (u/P)Pv|<= ||u/P||_1 ||Pv||_(oo) $.
If we apply this inequality to a sequence $(v_k) in ccD(RR^n) $ converging to $0$ in $ccS$ , we note that the sequence of integrals is infinitesimal.
For selecting $P$ , if $u$ has degree $<=m $ we can choose for example $P(x)=1+|x|^(2m+2n)$.

[elgiovo]

[Camillo]

Unfortunately I can't visualize langle and rangle and is very difficult to read

[elgiovo]

This is seen by observing that a given polynomial $p$ belongs to $L_("loc")^1 (RR)$, with $|p(x)|<=C (1+|x|^2)^(N/2)=C langle x rangle^(N) $ for some $C$ and some $N$. In fact, $|langle p, phi rangle|=|int p* phi " d"x|<=C int langle x rangle^(-2)" d"x *"sup"{langle x rangle^(N+2)|phi(x)|}<=C' * p_(N+2)(phi)$, with $phi in ccS$. $p_(M)(cdot)$ is a seminorm used in the definition of $ccS$.