泛函分析习题集.pdf

 §1.0.1 1). ℄ (a). A ∩ (B\C) = (A ∩ B)\(A ∩ C); (b). (A\B)\C = A\(B ∪ C); (c). A\(B\C) = (A\B) ∪ (A ∩ C); (d). (A ∪ B)\C = (A\C) ∪ (B\C). 2). f (g h) = (f g) h. 3). X,Y F : X → Y  G : Y → X G F = IX , F G = IY . §1.0.2 1). ℄ (a). (a, b) (0, 1) (b). R (−1, 1). 2). R 3). f (x), g(x) A (a). supx∈A {f (x) + g (x)} ≤ supx∈A f (x) + supx∈A g (x); (b). supx∈A {f (x) + g (x)} ≥ supx∈A f (x) + infx∈A g (x); (c). infx∈A {f (x) + g (x)} ≥ infx∈A f (x) + infx∈A g (x); (d). infx∈A {−f (x)} = − supx∈A f (x). §1.0.3 1). ℓ∞ ℄ ∞ ℓ = {xn } sup |xn | ∞ . n≥1 ℓ∞ ℓ∞ 2). 1 ≤ p ≤ q +∞ ℓp ⊂ ℓq . 2 n 3). Pn [a, b] n + 1 1, t, t , , t Pn [a, b] ℄ 2 n 4). P [a, b] , 1, t, t , , t , P [a, b] ℄ 5). C [a, b] . 6). X,Y K T : X → Y T N (T ) = {θ}. 2 3 7). P (x , y ), P (x , y ), P (x , y ) 1 1 1 2 2 2 3 3 3 Z = ax + by + c Z Pk Æ hk , k = 1, 2, 3. §1.0.4 1). X {xk }n k=1 n x − a x