diff --git a/src/fa/lc/hahn-banach.tex b/src/fa/lc/hahn-banach.tex index 20bd5fd..b86e8b8 100644 --- a/src/fa/lc/hahn-banach.tex +++ b/src/fa/lc/hahn-banach.tex @@ -163,3 +163,20 @@ is lower semicontinuous and Borel measurable by \autoref{proposition:semicontinuous-properties}. \end{proof} +\begin{proposition} +\label{proposition:closed-subspace-induction} + Let $E$ be a locally convex space over $K \in \RC$, $F \subset E$ be a closed subspace, and $y \in E \setminus F$, then $F + Ky$ is also a closed subspace. + + In particular, all finite-dimensional subspaces of $E$ are closed. +\end{proposition} +\begin{proof} + Since $F$ is closed and $E$ is locally convex, there exists a continuous seminorm $\rho: E \to [0, \infty)$ such that $\rho(F) = \bracsn{0}$ and $\rho(y) \ne 0$. By the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, there exists $\phi \in E^*$ such that $F \subset \ker(\phi)$ and $\dpn{y, \phi}{E} = 1$. In which case, $\phi$ induces an isomorphism + \[ + E \to \ker(\phi) \times K \quad x \mapsto (x - \dpn{x, \phi}{E}y, \dpn{x, \phi}{E}) + \] + + whose restriction to $F + Ky$ is an isomorphism onto $F \times K$. + + Let $\angles{x_\alpha + \lambda_\alpha y}_{\alpha \in A} \subset F + Ky$ be net convergent in $E$. For each $\alpha \in A$, $\dpn{x_\alpha + \lambda_\alpha y, \phi}{E} = \lambda_\alpha$. By continuity of $\phi$, $\angles{\lambda_\alpha}_{\alpha \in A}$ is convergent, and $\angles{x_{\alpha}}_{\alpha \in A}$ is also convergent. Since $\angles{x_\alpha}_{\alpha \in A} \subset F$ and $F$ is closed, $\lim_{\alpha \in A}x_\alpha \in F$. Therefore $\lim_{\alpha \in A}(x_\alpha + \lambda_\alpha y) \in F + Ky$, and $F + Ky$ is also a closed subspace. +\end{proof} + diff --git a/src/fa/lc/quotient.tex b/src/fa/lc/quotient.tex index 3aa02e7..ac73e78 100644 --- a/src/fa/lc/quotient.tex +++ b/src/fa/lc/quotient.tex @@ -24,6 +24,14 @@ As this holds for all $y \in x + M$ and $y' \in x' + M$, $\rho_M(x + x') \le \rho_M(x) + \rho_M(x')$. \end{proof} +\begin{lemma} +\label{lemma:quotient-norm-optimiser} + Let $E$ be a vector space over $K \in \RC$, $\rho: E \to [0, \infty)$ be a seminorm, $M \subset E$ be a vector subspace, $\rho_M: E/M \to [0, \infty)$ be its quotient by $M$, and $x \in E$ with $\rho_M(x) < 1$, then there exists $x_0 \in B_E(0, 1)$ such that $x + M = x_0 + M$. +\end{lemma} +\begin{proof} + Since $\inf_{y \in M}\rho(x - y) = \rho_M(x) < 1$, there exists $y \in M$ with $\rho(x - y) < 1$. Let $x_0 = x - y$, then $\rho(x_0) < 1$ as well and $x_0 + M = (x - y) + M = x + M$. +\end{proof} + \begin{definition}[Quotient Locally Convex Space] \label{definition:lc-quotient} Let $E$ be a locally convex space over $K \in \RC$ and $M \subset E$ be a vector subspace, then there exists $(\td E, \pi)$ such that: diff --git a/src/fa/norm/normed.tex b/src/fa/norm/normed.tex index 3238188..a79b18d 100644 --- a/src/fa/norm/normed.tex +++ b/src/fa/norm/normed.tex @@ -122,7 +122,7 @@ \begin{proposition} \label{proposition:dual-norm} - Let $E$ be a normed vector space, then for any $x \in E$, + Let $E$ be a normed vector space over $K \in \RC$, then for any $x \in E$, \[ \norm{x}_E = \sup_{\substack{\phi \in E^* \\ \norm{\phi}_{E^*} = 1}}\dpn{x, \phi}{E} \] @@ -132,3 +132,19 @@ \end{proof} +\begin{lemma}[Riesz] +\label{lemma:riesz-almost-orthogonal} + Let $E$ be an infinite-dimensional normed vector space over $K \in \RC$ and $\seq{\alpha_n} \subset (0, 1)$, then there exists $\seq{x_n} \subset \partial B_E(0, 1)$ such that + \[ + d(x_n, \text{span}\bracsn{x_j|1 \le j < n}) > \alpha_n \quad \forall n \in \natp + \] +\end{lemma} +\begin{proof} + Let $N \in \natp$ and suppose inductively that $\bracsn{x_n|1 \le n < N}$ have been constructed to satisfy the lemma for all $1 \le n < N$. Let $F_N = \text{span}\bracsn{x_n|1 \le n < N}$, then $F_N$ is finite-dimensional, and closed by \autoref{proposition:closed-subspace-induction}. As $E$ is infinite-dimensional, $E/F_N$ is non-trivial, and there exists $y_N \in E$ such that $\norm{y_N + F_N}_{E/F_N} > \alpha_N$. By \autoref{lemma:quotient-norm-optimiser}, there exists $z_N \in B_E(0, 1)$ such that $z_N + F_N = y_N + F_N$. As $z_N + F_N \ne 0$, $z_N \ne 0$ as well. Let $x_N = z_N/\norm{z_N}_{E} \in \partial B_E(0, 1)$, then + \[ + \norm{x_N + F_N}_{E/F} > \norm{z_N + F_N}_{E/F} = \norm{y_N + F_N}_{E/F} > \alpha_N + \] + + Thus $d(x_N, \text{span}\bracsn{x_n|1 \le n < N}) = \norm{x_N + F_N}_{E/F} > \alpha_N$. +\end{proof} + diff --git a/src/logic/index.tex b/src/logic/index.tex new file mode 100644 index 0000000..326a1db --- /dev/null +++ b/src/logic/index.tex @@ -0,0 +1,4 @@ +\part{Mathematical Logic} +\label{part:logic} + +This section is plagiarised from diff --git a/src/logic/tp/index.tex b/src/logic/tp/index.tex new file mode 100644 index 0000000..95d2955 --- /dev/null +++ b/src/logic/tp/index.tex @@ -0,0 +1,5 @@ +\chapter{Truth and Proof} +\label{chap:truth-and-proof} + +\input{./lang.tex} + diff --git a/src/logic/tp/lang.tex b/src/logic/tp/lang.tex new file mode 100644 index 0000000..5ae0778 --- /dev/null +++ b/src/logic/tp/lang.tex @@ -0,0 +1,15 @@ +\section{Languages} +\label{section:languages} + +\begin{definition}[Language] +\label{definition:language} + A \textbf{language} is a triplet of symbol sets $(\mathcal{F}, \mathcal{R}, \mathcal{C})$ and assignments $n_{\cdot}: \mathcal{F} \to \natp$ and $n_{\cdot}: \mathcal{R} \to \natp$, where: + \begin{enumerate} + \item For each $f \in \mathcal{F}$, $f$ is a function symbol in $n_{f}$ variables. + \item For each $R \in \mathcal{R}$, $R$ is a $n_R$-ary relation symbol. + \item For each $C \in \mathcal{C}$, $C$ is a constant symbol. + \end{enumerate} +\end{definition} + + +