Show that the vector space, $P_n$, of all the real polynomial functions of degree less than n, is a Banach Space for any norm define.
I think if I prove that $P_{n}$ is a Banach Space with the norm $|| |p| || = ||(a_0,a_1,...,a_n-1)||$, where $(a_0,a_1,...,a_{n-1})$ is a vector in $R^n$ wich coordinates are the coeficients of $p$ and $||.||$ is any norm in $R^n$, I can prove that is for any norm define in it.
Thank you
It's a finite dimensional real vector space, so all you need to show is that all norms are equivalent over $\mathbb R^n$ and any norm on $P_n$ will arise as a norm over $\mathbb R^n$ under the isomorphism $\phi : P_n \to \mathbb R^n$ you chose. If you want me to give you the details, ask for them. They can probably be found in your notes if you have any.
P.S. I think $P_n$ being the space of polynomials of degree less than $n$ includes degree $n$ polynomials, at least that's what I always see so I think of it as a standard.
Hope that helps,