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