Denote C([0,1],R) the real-valued continuous function defined on [0,1]. 1)Prove that C([0,1],R) is an integral domain. 2)Let x_0 lie in [0,1], Denote I(x_0)={f in C([0,1],R): f(x_0)=0} Prove that I(x_0) is a maximal ideal of C([0,1],R). 3)Find all it's maximal ideal. 4)Find all it's ideal. 5)We know that the quotient of an integral domain by a maximal ideal is isomorphic to a field. Prove that C([0,1],R)/I(x_0) is isomorphic to R. 6)(C([0,1],R),∥∥_∞) is Banach. 7)C([0,1],R)*(the dual space) is isomorphic to the space of functions of bounded variation on [0,1]. Note that BV[0,1] is a Banach space in which norm is defined by the total variation of the functions. 8)On C([0,1],R), we define 1 <f,g> = ∫ f(x)g(x)dx, 0 then prove that (C([0,1],R),<,>) is an inner product space and find its completion.(The completion of that is L^2([0,1])). 8-1)If f is in C([0,1],R), and 1 ∫ f(x)x^ndx=0 for all n, 0 then prove that f≡0 on [0,1]. 8-2)If f is in L^2([0,1]), 1 ∫ f(x)x^ndx=0 for all n, 0 then prove that f = 0 a.e on [0,1]. 8-3)Compute the Gram-Schmidt process of {1,x,x^2,...,} in C([-1,1],R). You can find the orthogonal polynomials are Legendre polynomials. You can also prove that the set of Legendre polynomials is a basis for (C([-1,1],R),<,>). You can also find the relation between the theory of function spaces and the theory of differential equations. We all know that the legendre polynomials satiesfy ((1-x^2)y')'+n(n+1)y = 0. y(-1)=y(1)=0. 8-4)You can also prove that {1.√2cos(2nπx),√2sin(2nπx)}_{n in |N} is an orthonomal basis for C([-1,1],R).The Fourier series related to the the theory of function spaces. 9) Let O([-1,1]) denote the odd continuous functions on [-1,1] and E([-1,1]) are even continuous functions on [-1,1], prove that C([-1,1]) = O([-1,1]) ⊕ E([-1,1]). 講了那麼多，主要是希望給準備數研所的同學一個方向。線代、高微、代數、微方 等等都是彼此相關聯的。你們可以藉由做題目中去尋找彼此的關連性。這些題目， 是我用印象給大家的，其中難免有一些疏漏請多包涵。但念數學有一個很重要的能力是 你必須學著去把這些看似不相關的觀念統整起來。我最後再舉兩個例子作為我這篇文章 的結語：如果考慮一個複矩陣A(n*n)，與線性微分系統 dx -- = Ax x(0)=v.(x 在 R^n). dt (A不見得可對角化) 則x(t) = exp(tA)v. 在這之中牽扯到幾個概念：exp(A)怎麼定義，收斂與否？ exp(tA)的計算就牽扯到Jordan form 的計算。 在(C[0,1],<,>)上我們定義算子A:C[0,1]->C[0,1]為 1 t A[f](x) = ∫ ∫ f(u)dudt x 0 那麼 A是定義在 C[0,1]上的線性算子。試證明 A is self-adjoint。 找出A 的eigenvalue以及找出他的spectral 分解。在解這個問題的同時， 就會利用到1)微分方程2)複便函數論(要求出cos(z)的無窮積)。 在解這個問題的過程中，spectral 分解中就內蘊了Fourier theory。 原本當初Fourier也是由此發展出富立業理論的。 由此可知，函數空間的重要性在哪。 --

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