COMPACT Flashcards
1
Q
T COMPACT ⟹
A
T BOUNDED
2
Q
T∈L(X,Y) COMPACT ⟺
A
(xn) bounded
⟹
T(xn) has convergent subsequence
3
Q
T COMPACT ⟸
A
T∈B(X,Y) and dim ran T < ∞
4
Q
SPECTRAL THEOREM FOR COMPACT OPERATORS
A
X banach and T∈K(X)
- ∀ε>0, the number of eigenvalues λ with |λ|>= ε is finite
- eigen value λ≠0 of T ⟹
dimker(T-λ)< ∞ - dimX= ∞ ⟹ 0 ∈ δ(T)
