Real Function is Linearly Dependent with Zero Function

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Theorem

Let $\map f x$ be a real function defined on a closed interval $\closedint a b$.

Let $\map g x$ be the constant zero function on $\closedint a b$:

$\forall x \in \closedint a b: \map g x = 0$


Then $f$ and $g$ are linearly dependent on $\closedint a b$.


Proof

We have that:

$\forall x \in \closedint a b: \map g x = 0 = 0 \times \map f x$

and $0 \in \R$.

Hence the result by definition of linearly dependent real functions.


Sources