課程名稱︰泛函分析
課程性質︰數學所選修
課程教師︰王振男
開課學院：理學院
開課系所︰數學所
考試日期（年月日）︰2017/06/12
考試時限（分鐘）：130分鐘，10:20-12:30。
試題 : （有些數學式用latex的表示式）
1. Let K : [0,1] x [0,1] \to \mathbb{R} be the characteristic function of {(x,
y): y \geq -x+1} and T be an integral operator on L^2([0,1]) with kernel K, i.
e., Tf(x)=\int_{1-x}^1 f(y) dy. Show that T is self-adjoint and compact. Furth
ermore, find all eigenvalues and eigenfunctions of T.
2. Let T : H \to H be a densely defined closed operator. Prove that H x H = VG
(T) \oplus G(T^*), where V{a,b}={-b,a} for a,b \in H.
3. Let C([0,1]) be the set of continuous functions on [0,1]. Let ||f||_1=\int_
0^1 |f(t)| dt. Is C([0,1],||‧||_1) a Banach space? Why?
4. Let V be a linear space. Assume that ||‧||_1 and ||‧||_2 are norms on V s
uch that (V,||‧||_1) and (V,||‧||_2) are Banach spaces. If there exists a co
nstant C_1 such that ||x||_1 \leq C_1 ||x||_2 for all x \in V, then there exis
ts some C_2 > 0 such that ||x||_2 \leq C_2 ||x||_1 for all x \in V.
5. Let K be a compact operator and {A_n} be a sequence of bounded operators on
a Hilbert space H. Assume that A_n \to A in the strong operator topology. Sho
w that KA_n \to KA in the norm topology.