E,F banach space, T $\in$ K(E,F) dim E= $\infty$ show R(T) is closed

Question

I was confuse about how to prove this ~~ My thought is prove the N(T) is finite dimension first then use this property to get R(T) is closed

I forgot to add one more assmption ~~~ it is bounded below
I know how to prove it ~~ thx for answer — Chun-yen Wu, Mar 09 '16 at 07:37

score 1 · Answer 1 · answered Mar 09 '16 at 07:27

1

This is wrong. $T:\ell^2\to\ell^2$, $(x_n)_{n\in\mathbb N} \mapsto (x_n/n)_{n\in\mathbb N}$ is compact with dense range.

answered Mar 09 '16 at 07:27

Jochen

1 Answers1