Is $\{f_{a} \mid a \in \mathbb{R}\}$ a basis for $\mathbb{R}^{\mathbb{R}}$.

Question

The question is given below:

I know the definition of the basis, it must be linearly independent and it must span the space, but I am stucked in applying the definition here, could anyone help me please in doing so?

Answer 1

To check linear independence we have to check that every finite subset of $\{f_a\mid a\in\mathbb{R}\}$ is linearly independent. To this end let $\{a_1,\dotsc, a_n\}$ be a collection of distinct real numbers and suppose that $$ c_1f_{a_1}+\dotsb+c_nf_{a_n}=0 $$ for some $c_i\in\mathbb{R}$ where the $0$ denotes the zero function. Evaluate both sides at $a_i$ to conclude that $c_i=0$.

For spanning, note that the span of $\{f_a\mid a\in\mathbb{R}\}$ equals the set of functions $E$ that are equal to zero except at finitely many points. In particular $f(x)=x$ is not in the span and $E\ne \mathbb{R}^{\mathbb{R}}$ whence $\{f_a\mid a\in\mathbb{R}\}$ is not a basis for $\mathbb{R}^{\mathbb{R}}$ over $\mathbb{R}$

Answer 2

No, it is not. It is a linearly independent set but you cannot write $f(x)=x$ as a finite linear combination of $f_a$'s.