In my classwork I have to provide examples of infinite dimensional vector spaces with metric generated by their norm. I readily provide space of continuous function $C[a, b]$ and Hilbert sequence space $\mathscr l^2$ with their respective norm as the easiest examples, but I am still looking for a few more simple examples.

From this old MSE link here I see that the function space $L^p$ and the polynomials in one variable $\mathbb R[x]$ are also simple examples. I know from other resources that the norm to the function space $L^p$ is

$$\lVert f \rVert_p = \left( \int^a_b \lvert f(x) \rvert^p dx \right)^{\frac{1}{p}}$$ which I think is capable of generating a metric by $ d(f, g) = \lVert f - g \rVert_p$.

And here is my question: Does anyone know of a norm to the above polynomials in one variable $\mathbb R [x]$ that is capable of generating a metric? Thanks for your time and effort.

  • 1
    $\begingroup$ Every norm generates a metric, so any norm on polynomials will do (and there are lots to choose from.) $\endgroup$ Commented Jun 12, 2015 at 22:28
  • $\begingroup$ Ah, thanks for pointing that out! Can you give me an example of the norm, even the simplest one? Thanks again. $\endgroup$
    – A.Magnus
    Commented Jun 12, 2015 at 22:37
  • 2
    $\begingroup$ You can identify $\mathbb R [x]$ with a subspace of all the real sequences (in which all the terms are zero excepting finitely many); in this subspace the simplest nom is $||p||= Max |a_n|$ where $a_n$ are the coefficients of the polynomial $p$. $\endgroup$
    – Piquito
    Commented Jun 12, 2015 at 23:05
  • $\begingroup$ Got it! Thank you very much! $\endgroup$
    – A.Magnus
    Commented Jun 13, 2015 at 12:37


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