中原大學89學年度數學學系博士班招生考試命題

私立中原大學八十九學年度博士班招生考試命題紙

所組別：

數學學系

科目：

泛涵分析

考試時間：

6月14日第2節

In problems 1-3.let X and Y be normed linear spaces and let L(X,Y)be the normed linear space of all bounded linear transformations of X into Y.

<r<1.Show that there exists a vector

such that

for all

1.	(20%) Let M be a closed proper linear subspace of X and let r be a real number such that 0<1. Show that there exists a vector u
2.	(15%) if Y is a Banach space, show that L(X,Y) is a Banach space.
3.	(15%) If X is finite dimensional,show that every linear operator on X is bounded.
4.	(20%) Let H be a Hilbert space, let M be a closed linear subspace of H and .show that
5.	(15%)Prove that the usual norm of ,cannot be obtained from an inner product.
6.	(15%) (a) Let C₀ be the normed linear space of all complex sequences such that as , with norm .Let. Define for. Show that is a bounded linear functional on c₀ (b) let be a complex sequence.If converges for all, show that

---END---