# Thread: Infinite dimensional spaces and other questions.

1. ## Infinite dimensional spaces and other questions.

thanks, these are some of my questions :

1-in our lecture notes i face this statement but i can't understand it :
" it can happen for infinite dimensional spaces , that 𝐸𝑖⊕ 𝐸𝑖⊥ ≠ E. example :let 𝐸𝑖 ⊂ 𝑙2 consist of vectors having only finitely many non-zero components :
then 𝐸1⊥={0} but 𝐸1 ≠ 𝑙2"
2- i understand the definition of closed set is complement of open set so intersection of them should be zero , but in Wikipedia for definition of closed set i found picture, that shown a ball , it says that interior of this ball is open set and interior and border is closed set , how it is possible ?

thanks

2. The characters you have in your first point in your last post are unintelligible. Could you possibly write them up in LaTeX? The \otimes and \perp commands might be useful.