COMPACT

Question 1

T COMPACT ⟹

Answer 1

T BOUNDED

Question 2

T∈L(X,Y) COMPACT ⟺

Answer 2

(xn) bounded
⟹
T(xn) has convergent subsequence

Question 3

T COMPACT ⟸

Answer 3

T∈B(X,Y) and dim ran T < ∞

Question 4

SPECTRAL THEOREM FOR COMPACT OPERATORS

Answer 4

X banach and T∈K(X)

1. ∀ε>0, the number of eigenvalues λ with |λ|>= ε is finite
2. eigen value λ≠0 of T ⟹
   dimker(T-λ)< ∞
3. dimX= ∞ ⟹ 0 ∈ δ(T)