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## Homework Statement

This is only part of a problem I am working on, but the only part that I have questions about is the following:

Show that [itex]\mathcal{F}(\mathbb{R})[/itex] is infinite dimensional.

## Homework Equations

[itex]\mathcal{F}(\mathbb{R})[/itex] is the set of all functions that map real numbers to real numbers.

[itex]f_n[/itex] is the function defined by the rule [itex]f_n(x) = e^{nx}[/itex] for [itex]n \in \mathbb{N}[/itex]

## The Attempt at a Solution

Suppose [itex] \mathcal{F}(\mathbb{R}) [/itex] is finite dimensional.

This means that there exists a finite basis for [itex] \mathcal{F}(\mathbb{R}) [/itex].

Consider the set of vectors [itex] \mathcal{E} = \{ f_1, f_2, \dotsc, f_n \} [/itex] for some [itex] n \in \mathbb{N} [/itex].

Suppose [itex] \mathcal{E} [/itex] is a basis for [itex] \mathcal{F}(\mathbb{R}) [/itex], and consider the vector [itex] f_{n+1} [/itex] in [itex] \mathcal{F}(\mathbb{R}) [/itex].

Since [itex] \mathcal{E} [/itex] spans [itex] \mathcal{F}(\mathbb{R}) [/itex], we see that [itex] f_{n+1} \in \text{span}\{\mathcal{E}\} [/itex].

This means that

[itex] f_{n+1} = \sum_{k=1}^n a_k f_k = a_1 f_1 + a_2 f_2 + \dotsb + a_n f_n [/itex]

for some [itex] a_1, a_2, \dotsc, a_n \in \mathbb{R} [/itex].

Equivalently, this means that

[itex] a_1 f_1 + a_2 f_2 + \dotsb + a_n f_n - f_{n+1} = 0. \quad \quad (1) [/itex]

It has been shown that [itex] \{ f_1, f_2, \dotsc, f_{n+1} \} [/itex] is linearly independent.

This means that each coefficient of [itex] f_i [/itex] equals 0 for [itex] i = 1, 2, \dotsc, n+1 [/itex].

But this is impossible, as [itex] -1 \neq 0 [/itex].

Hence, [itex] \mathcal{E} [/itex] does not span [itex] \mathcal{F}(\mathbb{R}) [/itex].

I feel like I have't shown that [itex]\mathcal{F}(\mathbb{R})[/itex] is infinite dimensional.

It seems like I have shown that *some* finite bases do not work for [itex]\mathcal{F}(\mathbb{R})[/itex].

I think that I am close, but I think there is something missing.

Could someone point me in the right direction to finish this proof?

Thank you!!!