Proposition 1

Tag 084Y, permanent. Cite it as [Bourbaki, Tag 084Y].

Functions of a Real Variable, chapter I, § 3. DERIVATIVES OF HIGHER ORDER

The set of vector functions defined on an interval $I \subset \mathbf{R}$ , taking values in a given topological vector space $E$ , and having an $n^{th}$ derivative on $I$ , is a vector space over $\mathbf{R}$ , and $f \mapsto D^n f$ is a linear mapping of this space into the vector space of linear mappings from $I$ into $E$ .

One proves the formulae

D^n(f + g) = D^n f + D^n g \tag{2}

D^n(fa) = D^n f . a \tag{3}

by induction on $n$ when $f$ and $g$ have an $n^{th}$ derivative on $I$ ( $a$ being constant).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

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