0

Can we say that every linear map on Banach space $X$, for example $ f : X ‎‎‎‎\longrightarrow ‎\mathbb{C}‎‎ $ must be continuous and bounded?

if the answer is negative, please give me an example.

suomynonA
  • 6,895
m.t
  • 19
  • Unbounded linear functionals can be constructed as open ball suggests, there is a more detailed discussion here: http://math.stackexchange.com/questions/99206/discontinuous-linear-functional – Andres Mejia Nov 27 '16 at 12:32

1 Answers1

5

If $X$ is infinite dimensional, no. In fact you can construct one given any such $X$. Let $\{e_a: a\in I \}$ be a basis of $X$ where $\|e_a\| = 1$ for all $a$. There exists a countable subset (now I'll be a little bit sloppy) $\{e_n :n\in \Bbb N\}$. Now define $L: X \to \Bbb C$ to be the linear function such that $Le_a = 0$ if $a\in I\setminus \Bbb N$ and $Le_n = n$ for $n\in \Bbb N$. Then $L$ is not bounded.

  • 1
    @IvoTerek thanks! I believe that sloppiness is a good thing when the details are just noise. –  Nov 27 '16 at 16:08