【归档】证明区间[0, 1]上的所有实值连续函数构成的实向量空间是无限维的

连续函数构成的实向量空间是无限维的"/>

【归档】证明区间[0, 1]上的所有实值连续函数构成的实向量空间是无限维的

Note: 旧的wordpress博客弃用，于是将以前的笔记搬运回来。

Question:
Show that the real vector space consisting of all real-valued continuous function on the interval [ 0 , 1 ] [0, 1] [0,1] is infinite-dimensional.
证明区间 [ 0 , 1 ] [0, 1] [0,1]上的所有实值连续函数构成的实向量空间是无限维的。
Solution:
Since we know the subspace of finite-dimensional vector space is a finite-dimensional vector space, if a space’s subspace is infinite-dimensional, this space will be infinite-dimensional.
Now think about P ( F ) P(\mathbb{F}) P(F) on the interval [0, 1]. Take a list of polynomials of the P ( F ) P(\mathbb{F}) P(F). Let m be the maximum power of this list of polynomials. Then the highest power of polynomials in the span of this polynomial list is m m m. Therefor the polynomials whose highest power is m + 1 m + 1 m+1 are not belong to this span. Thus there not exist any list can span P ( F ) P(\mathbb{F}) P(F). Hence P ( F ) P(\mathbb{F}) P(F) is infinite-dimensional.
Since p ( F ) p(\mathbb{F}) p(F) is a subspace of the vector space V (which is consists of all real-valued continuous functions on the interval [ 0 , 1 ] [0, 1] [0,1]), V is infinite-dimensional.