Merge pull request TikhonJelvis#161 from TikhonJelvis/appendix6-fix

quick fix

appendix6/appendix6.md

@@ -56,7 +56,7 @@ We represent function approximations by parameterized functions $f: \mathcal{X}
 
 #### $D[\mathbb{R}]$ as an Affine Space $\mathcal{P}$
 
-When performing Stochastic Gradient Descent or Batch Gradient Descent, parameters $\bm{p} \in D[\mathbb{R}]$ of a function approximation $f: \mathcal{X} \times D[\mathbb{R}] \rightarrow \mathbb{R}$ are updated using an appropriate linear combination of gradients of $f$ with respect to $w$ (at specific values of $x \in \mathcal{X}$). Hence, the parameters domain $D[\mathbb{R}]$ can be treated as an affine space (call it $\mathcal{P}$) whose associated vector space (over scalars field $\mathbb{R}$) is the set of gradients of $f$ with respect to parameters $\bm{p} \in D[\mathbb{R}]$ (denoted as $\nabla_{\bm{p}} f(x, \bm{p})$), evaluated at specific values of $x \in \mathcal{X}$, with addition operation defined as element-wise real-numbered addition and scalar multiplication operation defined as element-wise multiplication with real-numbered scalars. We refer to this Affine Space $\mathcal{P}$ as the *Parameters Space* and we refer to it's associated vector space (of gradients) as the *Gradient Space* $\mathcal{G}$. Since each *point* in $\mathcal{P}$ and each *translation* in $\mathcal{G}$ is an element in $D[\mathcal{R}]$, the $\oplus$ operation is element-wise real-numbered addition.
+When performing Stochastic Gradient Descent or Batch Gradient Descent, parameters $\bm{p} \in D[\mathbb{R}]$ of a function approximation $f: \mathcal{X} \times D[\mathbb{R}] \rightarrow \mathbb{R}$ are updated using an appropriate linear combination of gradients of $f$ with respect to $w$ (at specific values of $x \in \mathcal{X}$). Hence, the parameters domain $D[\mathbb{R}]$ can be treated as an affine space (call it $\mathcal{P}$) whose associated vector space (over scalars field $\mathbb{R}$) is the set of gradients of $f$ with respect to parameters $\bm{p} \in D[\mathbb{R}]$ (denoted as $\nabla_{\bm{p}} f(x, \bm{p})$), evaluated at specific values of $x \in \mathcal{X}$, with addition operation defined as element-wise real-numbered addition and scalar multiplication operation defined as element-wise multiplication with real-numbered scalars. We refer to this Affine Space $\mathcal{P}$ as the *Parameters Space* and we refer to it's associated vector space (of gradients) as the *Gradient Space* $\mathcal{G}$. Since each *point* in $\mathcal{P}$ and each *translation* in $\mathcal{G}$ is an element in $D[\mathbb{R}]$, the $\oplus$ operation is element-wise real-numbered addition.
 
 We define the gradient function
 $$G: \mathcal{X} \rightarrow (\mathcal{P} \rightarrow \mathcal{G})$$