課程名稱︰物理數學
課程性質︰選修
課程教師︰江衍偉
開課學院：電資學院
開課系所︰光電所
考試日期（年月日）︰2011/11/09
考試時限（分鐘）：2hr
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.
1 -1
For the natrix A = [ ] , find (a) its adjoint A* , (b)its null space
-6 6
N(A),(c) the null space of its adjoint,N(A*),(d)the pseudo-inverse A' of A ,
and (e) AA' , (f) compute A*AA' and explain the meaning of the result briefly.
(18%)
2.
(a) What is the Schwarz inequality ? Write down its expression and describe its
validity condition.
(b) By using the Schwarz inequality , show that L^2 [a,b] is a subset of
L^1 [a,b].
(c) Again by using the Schwarz inequality , show that if a function sequence
{f_n (x) } converges with respect to the L^2 norm , then if converges to
the same limit function with respect to the L^1 norm.(18%)
3.
(a) Why do we need to define a metric d in a function space? Explain briefly.
(b) The word "complete" has two different meaning in our textbook.
Please give appropriate definitions and explain briefly. (10%)
4.
Consider a linear function T defined by Tf=f(0) , with f∈C [-1,1].
(a) Is T a bounded functional for f measured with the L^1 norm ?
Explain briefly.
(b) Is T a bounded functional for f measured with L^2 norm ?
Explain briefly,
(c) Is T a bounded functional for f measured with uniform norm?
Explain briefly. (15%)
5.
1
(a) Solve the integral equation 5x^4＋2－u(x)+∫ (xξ^2－4)u(ξ)dξ=0
0
(b) Verify your solution by substituting it back in the integral equation.(15%)
6.
(a) If N is a null space of a bounded linear functional defined on Hilbert
space H and is a proper subset of H. Show that N',the orthogonal
complement of N, is a one-dimensional space.
(b) Discuss briefly the role that the Riesz representation theorem plays in
establishing the theory of distributions (12%)
7.
Evaluate
(a) lim cos[n^6 (x+3n^3)] and (b) lim n^15 cos[n^6(x+3n^3)], in the
n->∞ n->∞
sense of distribution. (12%)
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