definition of normed covectors (dual) space of normed vectors space
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 tensors space with respect to field and k vectors spaces and vectors space over field.
- The reader knows a definition of bounded map between normed vectors spaces.
- The reader admits the proposition that for any vectors space, any nonempty subset of the vectors space is a vectors subspace if and only if the subset is closed under linear combination.
Target Context
- The reader will have a definition of normed covectors (dual) space of normed vectors space.
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:
\( F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure (so, an \(F\) vectors space) and with the canonical norm, \(\vert \bullet \vert\)
\( V\): \(\in \{\text{ the } F \text{ vectors spaces }\}\), with any norm, \(\Vert \bullet \Vert\)
\(*V^*\): \(= \{v^* \in L (V: F) \vert v^* \in \{\text{ the bounded maps }\} \}\), \(\in \{\text{ the } F \text{ vectors spaces }\}\), with the norm specified below
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Conditions:
\(\forall v^* \in V^* (\Vert v^* \Vert = sup_{v \in V \setminus \{0\}} \vert v^* (v) \vert / \Vert v \Vert )\)
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This norm for \(V^*\) is called "operator norm".
2: Note
The canonical norm for \(\mathbb{R}\) or \(\mathbb{C}\) means taking the absolute value, which is indeed a norm: for each \(v_1, v_2 \in F\) and each \(r \in F\), 1) \(0 \le \vert v_1 \vert\) and \(0 = \vert v_1 \vert\) equals \(v_1 = 0\); 2) \(\vert r v_1 \vert = \vert r \vert \vert v_1 \vert\); 3) \(\vert v_1 + v_2 \vert \le \vert v_1 \vert + \vert v_2 \vert\).
\(\Vert v^* \Vert \in \mathbb{R}\), because \(v^*\) is bounded.
Let us see that \(V^*\) is indeed an \(F\) vectors space.
\(L (V: F)\) is a \(F\) vectors space, as is mentioned in Note for the definition of tensors space with respect to field and k vectors spaces and vectors space over field.
For each \(v^*_1, v^*_2 \in V^*\) and each \(r_1, r_2 \in F\), \(r_1 v^*_1 + r_2 v^*_2 \in V^*\), because while \(r_1 v^*_1 + r_2 v^*_2 \in L (V: F)\), \(r_1 v^*_1 + r_2 v^*_2\) is bounded, because \(sup_{v \in V \setminus \{0\}} \vert (r_1 v^*_1 + r_2 v^*_2) (v) \vert / \Vert v \Vert = sup_{v \in V \setminus \{0\}} \vert r_1 v^*_1 (v) + r_2 v^*_2 (v) \vert / \Vert v \Vert \le sup_{v \in V \setminus \{0\}} (\vert r_1 v^*_1 (v) \vert + \vert r_2 v^*_2 (v)) \vert / \Vert v \Vert \le sup_{v \in V \setminus \{0\}} \vert r_1 v^*_1 (v) \vert / \Vert v \Vert + sup_{v \in V \setminus \{0\}} \vert r_2 v^*_2 (v) \vert / \Vert v \Vert = sup_{v \in V \setminus \{0\}} \vert r_1 \vert \vert v^*_1 (v) \vert / \Vert v \Vert + sup_{v \in V \setminus \{0\}} \vert r_2 \vert \vert v^*_2 (v) \vert / \Vert v \Vert = \vert r_1 \vert sup_{v \in V \setminus \{0\}} \vert v^*_1 (v) \vert / \Vert v \Vert + \vert r_2 \vert sup_{v \in V \setminus \{0\}} \vert v^*_2 (v) \vert / \Vert v \Vert \lt \infty\).
So, \(V^*\) is an \(F\) vectors space, by the proposition that for any vectors space, any nonempty subset of the vectors space is a vectors subspace if and only if the subset is closed under linear combination: it is nonempty, because the \(0\) map is contained in it.
Let us see that the norm is indeed a norm.
Let \(v^*_1, v^*_2 \in V^*\) and \(r \in F\) be any.
1) (\(0 \le \Vert v^*_1 \Vert\)) \(\land\) (\((0 = \Vert v^*_1 \Vert) \iff (v^*_1 = 0)\)): \(0 \le sup_{v \in V \setminus \{0\}} \vert v^*_1 (v) \vert / \Vert v \Vert = \Vert v^*_1 \Vert\); if \(0 = \Vert v^*_1 \Vert\), \(sup_{v \in V \setminus \{0\}} \vert v^*_1 (v) \vert / \Vert v \Vert = 0\), which means that \(\vert v^*_1 (v) \vert = 0\) for each \(v \in V\), which means that \(v^*_1 = 0\); if \(v^*_1 = 0\), \(v^*_1 (v) = 0\) for each \(v \in V\), which means that \(\Vert v^*_1 \Vert = sup_{v \in V \setminus \{0\}} \vert v^*_1 (v) \vert / \Vert v \Vert = 0\).
2) \(\Vert r v^*_1 \Vert = \vert r \vert \Vert v^*_1 \Vert\): \(\Vert r v^*_1 \Vert = sup_{v \in V \setminus \{0\}} \vert r v^* (v) \vert / \Vert v \Vert = sup_{v \in V \setminus \{0\}} \vert r \vert \vert v^* (v) \vert / \Vert v \Vert = \vert r \vert sup_{v \in V \setminus \{0\}} \vert v^* (v) \vert / \Vert v \Vert = \vert r \vert \Vert v^*_1 \Vert\).
3) \(\Vert v^*_1 + v^*_2 \Vert \le \Vert v^*_1 \Vert + \Vert v^*_2 \Vert\): \(\Vert v^*_1 + v^*_2 \Vert = sup_{v \in V \setminus \{0\}} \vert (v^*_1 + v^*_2) (v) \vert / \Vert v \Vert = sup_{v \in V \setminus \{0\}} \vert v^*_1 (v) + v^*_2 (v) \vert / \Vert v \Vert \le sup_{v \in V \setminus \{0\}} (\vert v^*_1 (v) \vert + \vert v^*_2 (v)) \vert / \Vert v \Vert \le sup_{v \in V \setminus \{0\}} \vert v^*_1 (v) \vert / \Vert v \Vert + sup_{v \in V \setminus \{0\}} \vert v^*_2 (v) \vert / \Vert v \Vert = \Vert v^*_1 \Vert + \Vert v^*_2 \Vert\).
