Problem of a compact space

Question

Let $X$ be a compact $T_2$ space.Assume that the vector space of real-valued continuous functions on $X$ is finite dimensional.Show that $X$ is finite.

Spent nearly 3 hours on this problem.Cant figure out how to start.Hints required

Answer 1

It is known that every compact $T_2$ space is normal. We will prove that if $X$ is infinite, then there is an infinite set of linearly independent real-valued continuous functions on $X$.

Choose countably many points $\{x_n:n=0,1,2,\cdots\}$ in $X$ and choose an open neighborhood $U_n$ of $x_n$ which does not contain $x_i$ for all $i<n$. Since $X$ is normal, we can find an open set $V_n$ such that $x_n\in V_n\subset \overline{V_n}\subset U_n$. By Urysohn's lemma, we can find a continuous function $f_i$ s.t. $f_n=1$ over $\overline{V_n}$ and $f_n=0$ outside of $U_n$.

We will prove that $\{f_0,f_1,\cdots,f_n\}$ is linearly independent. Let $c_0f_0+\cdots c_nf_n=0$. Substitute $x=x_i$ then we get

$$f_0(x_i)\cdot c_0+f_1(x_i)\cdot c_1+\cdots+ f_i(x_i) \cdot c_i=0.$$

We do not know that $f_j(x_i)=0$ if $j<i$, but we know that $f_i(x_i)=1$. So we can make a homogeneous system of linear equations: $$\begin{bmatrix}1&0&0&\cdots&0 \\ f_0(x_1)&1&0&\cdots&0\\ f_0(x_2)&f_1(x_2)&1&\cdots&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots\\ f_0(x_n)&f_1(x_n)&f_2(x_n)&\cdots&1 \end{bmatrix}\begin{bmatrix}c_0\\c_1\\c_2\\\vdots\\c_n\end{bmatrix}=0$$

Since the matrix is lower triangular and its diagonal has no zero entry, the matrix is invertible. So we get $c_0=c_1=\cdots=c_n=0$.

From this fact, we can conclude that $\{f_n:n=0,1,2,\cdots\}$ is an infinite set of linearly independent real-valued continuous functions.