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.

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。

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.