A linear operator on $l_1$ to a separable Banach space.

Question

$l_1$ is our (scalar) sequence space such that its norm $\Vert (a_n)\Vert=\Sigma_{n=1}^{\infty}|a_n|$ is finite. Let $(e_n)$ be the sequence of unit vectors in $l_1$.

Given $X$, a separable Banach space, and $(x_n)$, a sequence in $B_X$ that is dense in $B_X$. Define a linear operator $Q:l_1\rightarrow X$ by $$Q(e_n)=x_n,$$ then it says that in the light of $l_1$-norm property $Q$ is well-defined, bounded. I cannot put my finger on it. How can I show $Q$ is bounded?

http://math.stackexchange.com/questions/1538920/every-separable-banach-space-is-isomorphic-to-ell-1-a-for-some-closed-a-subs/1540061#1540061 — Tomasz Kania, Jan 12 '17 at 11:23

score 5 · Answer 1 · edited Jan 12 '17 at 09:11

5

You have $$\| Q(x) \| \le \sum_{i=1}^\infty |x_n| \, \|Q(e_n)\| \le \sum_{i=1}^\infty |x_n| = \|x\|\ .$$

This shows that $Q$ is bounded.

edited Jan 12 '17 at 09:11

Duchamp Gérard H. E.

3,644

answered Jan 12 '17 at 08:32

gerw

33,373

Thank you! I originally bounded each $a_n$ so I was stuck :( – CSH Jan 12 '17 at 08:49

A linear operator on $l_1$ to a separable Banach space.

1 Answers1