Theorem 26.4.6: Power Rule | Jerry's Digital Garden

Theorem 26.4.6 (Power Rule).label Let $E$ be a topological vector space, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, $F$ be a separated locally convex space, $T \in B^{n}_{\sigma}(E; F)$.

For each $1 \le m \le n$, let $\text{Inj}([m]; [n])$ be the set of injective mappings from $[m]$ to $[n]$. For any $\phi \in \text{Inj}([m]; [n])$, denote $\phi^{c} \in \text{Inj}([n-m]; [n] \setminus \phi([m]))$ as the unique increasing injective map. For each $h \in E^{n-m}$, $k \in E^{m}$, and $1 \le j \le n$, write

\[(h, k)_{\phi} = \begin{cases}k_{\phi^{-1}(j)}&j \in \phi([m]) \\ h_{(\phi^c)^{-1}(j)}&j \not\in \phi([m])\end{cases}\]

For any $x \in E$, denote $x^{(m)}$ as the tuple of $x$ repeated $m$ times, then the mapping $f: E \to F$ defined by $x \mapsto T(x^{(n)})$ is infinitely $\tilde\sigma$-differentiable on $E$, where

(1)
For each $1 \le m \le n$, $x \in E$, and $h \in E^{m}$,
\[D^{m}_{\sigma} f(x)(h) = \sum_{\phi \in \text{Inj}([m]; [n])}T((x^{(n-m)}, h)_{\phi}]\]

In particular,
\[D^{n}_{\sigma} f(x)(h) = \sum_{\phi \in S_n}T(h_{\phi(1)}, \cdots, h_{\phi(n)})\]
(2)
For each $m > n$ and $x \in E$, $D^{m}_{\sigma} f(x) = 0$.

Notably, if $T \in L^{(n)}(E; F)$ is symmetric or $T \in L^{n}(E; F)$, then $T$ is infinitely $\sigma$-differentiable on $E$.

Proof. (1): Let $0 \le m \le n - 1$ and suppose inductively that (1) holds for $m$. For each $x, h \in E$, $S \subset [n-m]$, and $1 \le j \le n - m$,

\[[(x, h)_{S}]_{j} = \begin{cases}h &j \in S \\ x &j \not\in S\end{cases}\]

By the Binomial formula, for each $h \in E$ and $k \in E^{m}$,

\begin{align*}D^{m}_{\sigma} f(x + h)(k)&= \sum_{\phi \in \text{Inj}([m]; [n])}T[((x+h)^{(n-m)}, k)_{\phi}] \\&= \sum_{\phi \in \text{Inj}([m]; [n])}\sum_{S \subset [n-m]}T[((x, h)_{S}, k)_{\phi}]\end{align*}

For $\ell \ge 2$, maps in $B_{\sigma}^{\ell}(E; F)$ are $\sigma$-small, so

\[r(h) = \sum_{\phi \in \text{Inj}([m]; [n])}\sum_{\substack{S \subset [n-m] \\ |S| \ge 2}}T[((x, h)_{S}, k)_{\phi}]\]

is $\sigma$-small. Hence

\begin{align*}&D^{(m)}_{\sigma} f(x + h)(k) - D_{\sigma}^{(m)}f(x)(k) \\&= \sum_{\phi \in \text{Inj}([m]; [n])}\sum_{j = 1}^{n-m}T[((x, h)_{\bracs{j}}, k)_{\phi}] + r(h) \\&= \sum_{\phi \in \text{Inj}([m+1]; [n])}T[(x^{(n-m-1)}, (h, k))_{\phi}] + r(h)\end{align*}

and for any $h \in E^{m+1}$,

\[D_{\sigma}^{(m+1)}f(x)(h) = \sum_{\phi \in \text{Inj}([m+1]; [n])}T[(x^{(n-m-1)}, h)_{\phi}]\]

(2): By (1), $D^{n}_{\sigma} f$ is constant.$\square$

Jerry's Digital Garden

Direct Backlinks

Jerry's Digital Garden

Direct Backlinks