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

Question

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?

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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.

Answer 1

My comments as an answer.

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.

On an unrelated note, Amann–Escher is (very) often overly formal. A totally sufficient and easy-to-read proof would be the following:

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 $k$.