definition of derivative of map from open subset of normed vectors space into subset of normed vectors space at point
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of normed vectors space.
- The reader knows a definition of metric induced by norm on real or complex vectors space.
- The reader knows a definition of topology induced by metric.
- The reader knows a definition of linear map.
Target Context
- The reader will have a definition of derivative of map from open subset of normed vectors space into subset of normed vectors space at point.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( V_1\): \(\in \{\text{ the normed vectors spaces }\}\) with the topology induced by the metric induced by the norm
\( V_2\): \(\in \{\text{ the normed vectors spaces }\}\)
\( U_1\): \(\in \{\text{ the open subsets of } V_1\}\)
\( S_2\): \(\in \{\text{ the subsets of } V_2\}\)
\( f\): \(: U_1 \to S_2\)
\( u_1\): \(\in U_1\)
\(*{D f}_{u_1}\): \(: V_1 \to V_2\), \(\in \{\text{ the linear maps }\}\)
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Conditions:
\(\forall u_1 + u'_1 \in U_1 (f (u_1 + u'_1) = f (u_1) + {D f}_{u_1} (u'_1) + r (u_1, u'_1))\) where \(\lim_{\Vert u'_1 \Vert \to 0} \Vert r (u_1, u'_1) \Vert / \Vert u'_1 \Vert = 0\)
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2: Note
\({D f}_{u_1}\) does not necessarily exist.
Let us see that when \({D f}_{u_1}\) exists, it is unique.
Let us suppose that there was any other derivative, \(g: V_1 \to V_2\).
\(f (u_1 + u'_1) = f (u_1) + g (u'_1) + r' (u_1, u'_1)\) where \(\lim_{\Vert u'_1 \Vert \to 0} \Vert r' (u_1, u'_1) \Vert / \Vert u'_1 \Vert = 0\).
\({D f}_{u_1} (u'_1) - g (u'_1) + r (u_1, u'_1) - r' (u_1, u'_1) = 0\).
\(\Vert {D f}_{u_1} (u'_1) - g (u'_1) \Vert = \Vert {D f}_{u_1} (u'_1) - g (u'_1) + r (u_1, u'_1) - r' (u_1, u'_1) - (r (u_1, u'_1) - r' (u_1, u'_1)) \Vert = \Vert 0 - (r (u_1, u'_1) - r' (u_1, u'_1)) \Vert = \Vert (r (u_1, u'_1) - r' (u_1, u'_1)) \Vert \le \Vert r (u_1, u'_1) \Vert + \Vert r' (u_1, u'_1) \Vert\).
\(\lim_{\Vert u'_1 \Vert \to 0} \Vert {D f}_{u_1} (u'_1) - g (u'_1) \Vert / \Vert u'_1 \Vert \le \lim_{\Vert u'_1 \Vert \to 0} (\Vert r (u_1, u'_1) \Vert + \Vert r' (u_1, u'_1) \Vert) / \Vert u'_1 \Vert = \lim_{\Vert u'_1 \Vert \to 0} \Vert r (u_1, u'_1) \Vert / \Vert u'_1 \Vert + \lim_{\Vert u'_1 \Vert \to 0} \Vert r' (u_1, u'_1) \Vert) / \Vert u'_1 \Vert = 0 + 0 = 0\).
\({D f}_{u_1} (u'_1) - g (u'_1)\) is linear with respect to \(u'_1\), so, \({D f}_{u_1} (r u'_1) - g (r u'_1) = r ({D f}_{u_1} (u'_1) - g (u'_1))\), and \(\lim_{r \to 0} \Vert {D f}_{u_1} (r u'_1) - g (r u'_1) \Vert / \Vert r u'_1 \Vert = \lim_{r \to 0} \Vert r ({D f}_{u_1} (u'_1) - g (u'_1)) \Vert / (\vert r \vert \Vert u'_1 \Vert) = \lim_{r \to 0} \vert r \vert \Vert {D f}_{u_1} (u'_1) - g (u'_1) \Vert / (\vert r \vert \Vert u'_1 \Vert) = \lim_{r \to 0} \Vert {D f}_{u_1} (u'_1) - g (u'_1) \Vert / \Vert u'_1 \Vert = 0\), so, \(\Vert {D f}_{u_1} (u'_1) - g (u'_1) \Vert = 0\), which implies that \({D f}_{u_1} (u'_1) - g (u'_1) = 0\).
Although that holds only for each \(u'_1 \in V_1\) such that \(u_1 + u'_1 \in U_1\), for each \(u''_1 \in V_1\), as \(U_1\) is open, there is a small enough \(r \in \mathbb{R}\) such that \(0 \lt r\) and \(u_1 + r u''_1 \in U_1\), so, \({D f}_{u_1} (r u''_1) - g (r u''_1) = 0\), but the left hand side is \(r ({D f}_{u_1} (u''_1) - g (u''_1))\), so, \({D f}_{u_1} (u''_1) - g (u''_1) = 0\).
So, \({D f}_{u_1} = g\).
