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 $\struct {\map C I, +, \, \cdot \,}_\R$ be the continuous on closed interval real function vector space.
Then $\struct {\map C I, +, \, \cdot \,}_\R$ is not finite dimensional.
Proof
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 $n \le d$.
Aiming for a contradiction, suppose 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.
$\Box$
$\struct {\map C I, +, \, \cdot \,}_\R$ is not finite dimensional
Aiming for a contradiction, suppose $\struct {\map C I, +, \, \cdot \,}_\R$ 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 $n \in \N_{>0}$ and $n \le d$ is a basis for $\struct {\map C I, +, \, \cdot \,}_\R$.
The constant function $\map x t = 1$ belongs to $\map C I$.
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, $\struct {\map C I, +, \, \cdot \,}_\R$ is not finite dimensional.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $1.1$: Normed and Banach spaces. Vector Spaces