fixed lemma about orthogonal projection

inputs/lecture_14.tex

@@ -101,7 +101,7 @@ and then do the harder proof.
     i.e.~$H$ is a vector space with an inner product $\langle \cdot, \cdot \rangle_H$ which defines a norm by $\|x\|_H^2 = \langle x, x\rangle_H$
     making $H$ a complete metric space.
 
-    For any $x \in H$ and $K \subseteq H$ closed,
+    For any $x \in H$ and $K \subseteq H$ closed and convex,
     there exists a unique $z \in K$ such that the following equivalent conditions hold:
     \begin{enumerate}[(a)]
         \item  $\forall  y \in K : \langle x-z, y\rangle_H = 0$,

inputs/lecture_19.tex

@@ -148,6 +148,7 @@ However, some subsets can be easily described, e.g.
 \end{proof}
 
 \begin{theorem}
+    \label{thm:l1iffuip}
     Assume that $X_n \in L^1$ for all $n$ and $X \in L^1$.
     Then the following are equivalent:
     \begin{enumerate}[(1)]
@@ -189,7 +190,7 @@ However, some subsets can be easily described, e.g.
     (1) $\implies$ (2)
 
     $X_n \xrightarrow{L^1}  X \implies X_n \xrightarrow{\bP} X$
-    by Markov's inequality.
+    by Markov's inequality (see \autoref{claim:convimpll1p}).
 
     Fix $\epsilon > 0$.
     We have

inputs/prerequisites.tex

@@ -108,7 +108,8 @@ from the lecture on stochastic.
         Hence $\bE[|X_n - X|^q] \xrightarrow{n\to \infty} 0 \implies \bE[|X_n - X|^p] \xrightarrow{n\to \infty} 0$.
     \end{subproof}
     \begin{claim}
-    $X_n \xrightarrow{L^1} X \implies X_n\xrightarrow{\bP} X$
+        \label{claim:convimpll1p}
+        $X_n \xrightarrow{L^1} X \implies X_n\xrightarrow{\bP} X$.
     \end{claim}
     \begin{subproof}
         Suppose $\bE[|X_n - X|] \to  0$.
@@ -151,7 +152,9 @@ from the lecture on stochastic.
     \end{subproof}
     \begin{claim}
         \label{claim:convimplpl1}
-    $X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$
+        $X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$.%
+        \footnote{Note that the implication holds under certain assumptions,
+            see \autoref{thm:l1iffuip}.}
     \end{claim}
     \begin{subproof}
     Take $([0,1], \cB([0,1 ]), \lambda)$