# Amann/Escher, Analysis I, Exercise I.12.12: Prove that $\triangle_{h}^{k} \in \operatorname{Hom}\left(\mathbb{K}_{m}[X], \mathbb{K}_{m-k}[X]\right)$

I'm doing Problem I.12.12 from textbook Analysis I by Amann/Escher. Here

$$\mathbb{K}$$ denotes either of fields $$\mathbb{R}$$ and $$\mathbb{C}$$.

$$\mathbb{K}_{m}[X]$$ is the vector space of polynomials whose degree is less than or equal to $$m$$.

The difference operator $$\triangle^k_h$$ is defined by $$\triangle^k_h := \underbrace{\triangle_h \cdots \triangle_h}_{k \text{ times}}$$ where $$\triangle_h (p) := \dfrac{\sum_{i=0}^m p_i (X+h)^i - \sum_{i=0}^m p_i X^i}{h}, \quad p = \sum_{i=0}^m p_i X^i \in K_m[X]$$

Could you please verify whether my attempt is fine or contains logical gaps/mistakes?

My attempt

Lemma 1: If $$p \in \mathbb{K}_{m}[X]$$ then $$\triangle^k_h (p) \in \mathbb{K}_{m-k}[X]$$.

Lemma 2: $$\triangle_h (\alpha p) = \alpha (\triangle_h (p))$$ for all $$\alpha \in \mathbb K$$ and $$p \in \mathbb{K}[X]$$.

I proceed to prove the statement by induction on $$k$$. The statement trivially holds for $$k=0$$. Let it hold for $$k$$. Then $$\triangle_{h}^{k} \in \operatorname{Hom}\left(\mathbb{K}_{m}[X], \mathbb{K}_{m-k}[X]\right)$$. By lemma 1, we have $$\triangle^{k+1}_h \in {\left ( \mathbb{K}_{m-(k+1)}[X] \right )}^{\mathbb{K}_{m}[X]}$$. Next we prove that $$\triangle^{k+1}_h$$ is a vector space homomorphism. For $$\alpha, \beta \in \mathbb K$$ and $$p,q \in \mathbb{K}_{m}[X]$$, we have

\begin{aligned}\triangle^{k+1}_h (\alpha p + \beta q) &= \triangle_h \triangle^k_h (\alpha p + \beta q)\\ &=\triangle_h (\alpha \triangle^k_h (p) + \beta \triangle^k_h (q)), \quad \text{by inductive hypothesis} \\ &= \alpha (\triangle_h \triangle^k_h (p)) + \beta (\triangle_h \triangle^k_h (q)), \quad \text{by lemma 2} \\ &= \alpha \triangle^{k+1}_h (p) + \beta \triangle^{k+1}_h (q) \end{aligned}

This completes the proof.

• You use that $Δ_h$ is additive, refering to Lemma 2, yet it only claims that $Δ_h$ is homothetic (that is, it commutes with scalars). Jul 24 '19 at 16:23
• Hi @k.stm, you are totally correct. I implicit use the additive property without explicitly mentioning it. Beside that point, have you seen any other error? Jul 24 '19 at 16:28
• On an unrelated note, Amann–Escher is (very) often overly formal. A totally sufficient and easy-to-read proof would be “It’s easy to see that $Δ_h$ is linear and that for all nonconstant polynomials $f ∈ K[X]$, $\deg Δ_h(f) ≤ \deg f - 1$, whereas the constant polynomials are in the kernel of $Δ_h$. Since $Δ_h$ is linear, this can be checked on monomials. By induction $Δ_h^k$ is linear as well for all $k$ and reduces the degree of nonconstant polynomials by at least $k$.” Jul 24 '19 at 16:28
• Thank you so much for your detailed comment @k.stm. It would be great if you post it as an answer. Jul 24 '19 at 16:31

You use that $$Δ_h$$ is additive, refering to Lemma 2, yet it only claims that $$Δ_h$$ is homothetic (that is, it commutes with scalars). Other than that, it looks good.
It’s easy to see that $$Δ_h$$ is linear and further that for all nonconstant polynomials $$f∈K[X]$$, $$\deg Δ_h(f) ≤ \deg f − 1$$, whereas the constant polynomials are in the kernel of $$Δ_h$$. Since $$Δ_h$$ is linear, this can be checked on monomials. By induction $$Δ^k_h$$ is linear as well for all $$k ∈ ℕ$$ and reduces the degree of nonconstant polynomials by at least $$k$$.