Vector Space of Continuous on Closed Interval Real Functions is not Finite Dimensional

Theorem
Let $I := \closedint 0 1$ be a closed real interval.

Let $\struct {\map \CC I, +, \, \cdot \,}_\R$ be the continuous on closed interval real function vector space.

Then $\struct {\map \CC I, +, \, \cdot \,}_\R$ is not finite dimensional.

Monomials are linearly independent
Let $d \in \N_{>0}$.

Consider the set of real monomials of the following form:


 * $\map {x_n} t = t^n$

where $n \in \N_{>0}$ and $n \le d$.

the set of $x_n$ is not linearly independent.

Then:


 * $\forall n \in \N_{>0}: n \le d: \exists \alpha_n \in \R: \neg \forall n: \alpha_n \ne 0$

and:


 * $\ds \sum_{k \mathop = 1}^d \alpha_k t^k = 0$

Let $m \in \N_{>0}: m \le d$ be the smallest index such that $\alpha_m \ne 0$.

Then:


 * $\ds \forall t \in \closedint 0 1: \sum_{k \mathop = m}^d \alpha_k t^k = 0$

or


 * $\ds \forall t \in \hointl 0 1: \sum_{k \mathop = m}^d \alpha_k t^{k - d} = 0$

Note that:


 * $\ds \forall n \in \N_{>0}: \exists t \in \hointl 0 1: t = \frac 1 n$

Thus:


 * $\ds \forall n \in \N_{>0}: \sum_{k \mathop = m}^d \frac {\alpha_k} {n^{d - k} } = 0$

Passing the limit $n \to \infty$ gives us $\alpha_d = 0$.

This is a contradiction.

Hence, the set of $x_n$ is linearly independent.

$\struct {\map \CC I, +, \, \cdot \,}_\R$ is not finite dimensional
$\struct {\map \CC I, +, \, \cdot \,}_\R$ is finite dimensional and has the dimension $d$.

Any independent set of cardinality $d$ in a $d$-dimensional vector space is a basis for this vector space.

Then the set of monomials $x_n$ with $n \in \N_{\mathcal > 0}$ and $n \le d$ is a basis for $\struct {\map \CC I, +, \, \cdot \,}_\R$.

The constant function $\map x t = 1$ belongs to $\map \CC I$.

Then:


 * $\ds \forall n \in \N_{>0}: n \le d: \exists \beta_n \in \R: 1 = \sum_{k \mathop = 1}^d \beta_k \map {x_k} t$

Let $t = 0$.

Then $1 = 0$.

This is a contradiction.

Hence, $\struct {\map \CC I, +, \, \cdot \,}_\R$ is not finite dimensional.