1

Let $X$ be the vector space of all sequences in $\mathbb R$. Is it possible to define a norm on $X$ for which each of the co-ordinate functionals given by $f_n((x_m))=x_n$ is continuous. We know that it is true in $\ell^{p}(1\leq p<\infty)$. Can we expect the same for $X$ as well? Any hint will be appreciated.

Anupam
  • 4,908
  • 1
    This seems like a relevant post - https://math.stackexchange.com/questions/3164543/is-the-space-of-real-sequences-normable – Calvin Khor Oct 20 '21 at 07:31

1 Answers1

2

Suppose that such norm exists. Then for any $i\in\mathbb{N} $ there exists $c_i >0$ such that for all $x\in X$ we have $$||x||\geq c_i |x_i |$$ hence for all $x\in X$ we have $$||x||\geq \sum_{j=1}^{\infty } 2^{-i} c_i|x_i |.$$ Now take $x^0= (x^0_n )$ where $x^0_n = 2^{n} c_n^{-1}$ then $$||x^0 ||\geq \sum_{j=1}^{\infty} 1=\infty$$ and we obtain contradiction.