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]). 讲了那麽多，主要是希望给准备数研所的同学一个方向。线代、高微、代数、微方 等等都是彼此相关联的。你们可以藉由做题目中去寻找彼此的关连性。这些题目， 是我用印象给大家的，其中难免有一些疏漏请多包涵。但念数学有一个很重要的能力是 你必须学着去把这些看似不相关的观念统整起来。我最後再举两个例子作为我这篇文章 的结语：如果考虑一个复矩阵A(n*n)，与线性微分系统 dx -- = Ax x(0)=v.(x 在 R^n). dt (A不见得可对角化) 则x(t) = exp(tA)v. 在这之中牵扯到几个概念：exp(A)怎麽定义，收敛与否？ exp(tA)的计算就牵扯到Jordan form 的计算。 在(C[0,1],<,>)上我们定义算子A:C[0,1]->C[0,1]为 1 t A[f](x) = ∫ ∫ f(u)dudt x 0 那麽 A是定义在 C[0,1]上的线性算子。试证明 A is self-adjoint。 找出A 的eigenvalue以及找出他的spectral 分解。在解这个问题的同时， 就会利用到1)微分方程2)复便函数论(要求出cos(z)的无穷积)。 在解这个问题的过程中，spectral 分解中就内蕴了Fourier theory。 原本当初Fourier也是由此发展出富立业理论的。 由此可知，函数空间的重要性在哪。 --

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