## 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。

Question:

Show that the real vector space consisting of all real-valued continuous function on the interval $$[0, 1]$$ is infinite-dimensional.

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 $$P(\mathbb{F})$$ on the interval [0, 1]. Take a list of polynomials of the $$P(\mathbb{F})$$. 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 $$m$$. Therefor the polynomials whose highest power is $$m + 1$$ are not belong to this span. Thus there not exist any list can span $$P(\mathbb{F})$$. Hence $$P(\mathbb{F})$$ is infinite-dimensional.

Since $$p(\mathbb{F})$$ is a subspace of the vector space V (which is consists of all real-valued continuous functions on the interval $$[0, 1]$$), V is infinite-dimensional.