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. --

※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 101.14.182.130 ※ 文章網址: https://webptt.com/m.aspx?n=bbs/NTU-Exam/M.1498562063.A.1F2.html ※ 編輯: sheng0321 (49.215.200.84), 06/27/2017 22:23:54 ※ 編輯: sheng0321 (49.214.150.192), 06/28/2017 19:42:25 ※ 編輯: sheng0321 (101.15.17.211), 07/09/2017 13:56:56