From 62a8e78dfeb61c6c8e38aee069002b113a141919 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 15 Mar 2026 12:32:31 -0400 Subject: [PATCH] Typo fixes. --- src/dg/derivative/higher.tex | 32 +++++++++++++++++++++----------- src/topology/main/c0.tex | 4 ++-- src/topology/uniform/metric.tex | 11 ++++++----- 3 files changed, 29 insertions(+), 18 deletions(-) diff --git a/src/dg/derivative/higher.tex b/src/dg/derivative/higher.tex index 5f16fb7..832b735 100644 --- a/src/dg/derivative/higher.tex +++ b/src/dg/derivative/higher.tex @@ -35,26 +35,33 @@ A(h, k) &= Df(x + h)(k) + Df(x)(k) \\ &+ [f(x + h + k) - f(x + h) - Df(x + h)(k)] \\ &- [f(x + k) - f(x) - Df(x)(k)] \\ - &= D^2f(x)(h, k) + r_1(h) \cdot Df(x)(k) + &= D^2f(x)(h, k) + r_1(h) \cdot Df(x)(k) \\ &+ [f(x + h + k) - f(x + h) - Df(x + h)(k)] \\ &- [f(x + k) - f(x) - Df(x)(k)] \\ \end{align*} Let $B_h: B_E(0, r) \to F$ be defined by - \[ - B_h(k) = f(x + h + k) - f(x + k) - Df(x + h)(k) + Df(x)(k) - \] + \begin{align*} + B_h(k) &= f(x + h + k) - f(x + k) \\ + &- Df(x + h)(k) + Df(x)(k) + \end{align*} + then - \[ - B_h(k) - B_h(0) = f(x + h + k) - f(x + k) - Df(x + h)(k) + Df(x)(k) -f(x + h) + f(x) - \] + \begin{align*} + B_h(k) - B_h(0) &= f(x + h + k) - f(x + k) \\ + &- Df(x + h)(k) + Df(x)(k) \\ + &-f(x + h) + f(x) + \end{align*} + Now, there exists $r_2, r_3 \in \mathcal{R}_{B(E)}$ such that for any $k \in B(0, r)$, \begin{align*} DB_h(k) &= Df(x + h + k) - Df(x + k) - Df(x + h) + Df(x) \\ - &= D^2f(x)(h + k) + Df(x) - D^2f(x)(h) - Df(x) - D^2f(x)(k) + r_2(k) + r_3(h) \\ + &= D^2f(x)(h + k) + Df(x) - D^2f(x)(h) \\ + &- Df(x) - D^2f(x)(k) + r_2(k) + r_3(h) \\ &=r_2(k) + r_3(h) \end{align*} + By the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem}, \[ \norm{B_h(k) - B_h(0)}_F \le \norm{k}_E \cdot o(\norm{k}_E + \norm{h}_E) @@ -68,9 +75,12 @@ so $D^2f(x)(h, k) - D^2f(x)(k, h) = 0$. Now suppose that the proposition holds for $n$. Identify $L^n(E; F) = L^{2}(E; L^{n-2}(E; F))$, then for any $\seqf[n]{x_j} \subset E$, - \[ - Df(x)(x_1, \cdots, x_n) = Df(x)(x_1, x_2)(x_3, \cdots, x_n) = Df(x)(x_2, x_1)(x_3, \cdots, x_n) = Df(x)(x_2, x_1, x_3, \cdots, x_n) - \] + \begin{align*} + Df(x)(x_1, \cdots, x_n) &= Df(x)(x_1, x_2)(x_3, \cdots, x_n) \\ + &= Df(x)(x_2, x_1)(x_3, \cdots, x_n) \\ + &= Df(x)(x_2, x_1, x_3, \cdots, x_n) + \end{align*} + Since any element $\sigma \in S_n$ that does not fix $x_1$ is the composition of the transposition $(12)$ and an element that fixes $x_1$, $Df(x)$ is symmetric. \end{proof} diff --git a/src/topology/main/c0.tex b/src/topology/main/c0.tex index aee2e2c..74b6d9d 100644 --- a/src/topology/main/c0.tex +++ b/src/topology/main/c0.tex @@ -5,7 +5,7 @@ \label{definition:vanish-at-infinity} Let $X$ be a topological space, $E$ be a TVS over $K \in \RC$, and $f \in C(X; E)$, then $f$ \textbf{vanishes at infinity} if for every $U \in \cn_E^o(0)$, $\bracs{f \not\in U}$ is compact. - The set $C_0(X; \complex)$ is the space of all functions that vanish at infinity. + The set $C_0(X; E)$ is the space of all functions that vanish at infinity. \end{definition} \begin{proposition} @@ -35,6 +35,6 @@ \[ (\phi f - f)(X) = \underbrace{(\phi f - f)(\bracs{f \not\in U})}_{0} + \underbrace{(\phi f - f)(\bracs{f \in U})}_{\in U} \in U \] - + so $f \in \ol{C_c(X; E)}$. \end{proof} diff --git a/src/topology/uniform/metric.tex b/src/topology/uniform/metric.tex index aae8b57..d52b3bc 100644 --- a/src/topology/uniform/metric.tex +++ b/src/topology/uniform/metric.tex @@ -70,15 +70,16 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa \begin{enumerate} \item[(FB1)] For any $J, J' \subset I$ finite and $r, r' > 0$, \[ - \bigcap_{j \in J \cup J'}E(d_j, \min(r,r')E(d_j, r \wedge r') \subset \paren{\bigcap_{j \in J}E(d_j, r)} \cap \paren{\bigcap_{j \in J'}E(d_j, r')} + \bigcap_{j \in J \cup J'}E(d_j, r \wedge r') \subset \paren{\bigcap_{j \in J}E(d_j, r)} \cap \paren{\bigcap_{j \in J'}E(d_j, r')} \] \item[(UB1)] For each $i \in I$, $d(x, x) = 0$ for all $x \in X$. Thus for any $i \in I$ and $r > 0$, $E(d_i, r)$ contains the diagonal. \item[(UB2)] For each $J \subset I$ finite and $r > 0$, - \[ - \paren{\bigcap_{j \in J}E(d_j, r/2)} \circ \paren{\bigcap_{j \in J}E(d_j, r)} \subset \bigcap_{j \in J}E(d_j, r/2) \circ E(d_j, r/2) \subset \bigcap_{j \in J}E(d_j, r) - \] - + \begin{align*} + \paren{\bigcap_{j \in J}E(d_j, r/2)} \circ \paren{\bigcap_{j \in J}E(d_j, r)} &\subset \bigcap_{j \in J}E(d_j, r/2) \circ E(d_j, r/2) \\ + &\subset \bigcap_{j \in J}E(d_j, r) + \end{align*} + by the triangle inequality. \end{enumerate}