definition of \(L^p\) over measure space
Topics
About: measure space
The table of contents of this article
Starting Context
- The reader knows a definition of \(\mathcal{L}^p\) over measure space.
- The reader knows a definition of quotient vectors space of vectors space by vectors subspace.
- 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.
- The reader admits the proposition that for any set, any subset minus the intersection of any bunch of subsets is the union of the subset minus the bunch of subsets.
- The reader admits the proposition that for any set, the intersection of the union of any possibly uncountable number of subsets and any subset is the union of the intersections of each of the subsets and the latter subset.
Target Context
- The reader will have a definition of \(L^p\) over measure 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:
\( (M, A, \mu)\): \(\in \{\text{ the measure spaces }\}\)
\( \mathbb{C}\): \(= \text{ the complex Euclidean topological space }\) with the Borel \(\sigma\)-algebra
\( \mathbb{R}\): \(= \text{ the Euclidean topological space }\) with the Borel \(\sigma\)-algebra
\( F\): \(\in \{\mathbb{C}, \mathbb{R}\}\)
\( p\): \(\in \mathbb{R}\) such that \(1 \le p \lt \infty\)
\( \mathcal{L}^p (M, A, \mu, F)\):
\(*L^p (M, A, \mu, F)\): \(\mathcal{L}^p (M, A, \mu, F) / \{f \in \mathcal{L}^p (M, A, \mu, F) \vert \Vert f \Vert = 0\}\), \(= \text{ the quotient vectors space }\), with the norm specified below
\( \mathcal{L}^\infty (M, A, \mu, F)\):
\(*L^\infty (M, A, \mu, F)\): \(\mathcal{L}^\infty (M, A, \mu, F) / \{f \in \mathcal{L}^\infty (M, A, \mu, F) \vert \Vert f \Vert = 0\}\), \(= \text{ the quotient vectors space }\), with the norm specified below
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Conditions:
\(\Vert [f] \Vert = \Vert f \Vert\)
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2: Note
This is different from \(\mathcal{L}^p (M, A, \mu, F)\).
Let us see that \(\{f \in \mathcal{L}^p (M, A, \mu, F) \vert \Vert f \Vert = 0\}\) is indeed a vectors subspace of \(\mathcal{L}^p (M, A, \mu, F)\).
Let \(f_1, f_2 \in \{f \in \mathcal{L}^p (M, A, \mu, F) \vert \Vert f \Vert = 0\}\) and \(r_1, r_2 \in F\) be any.
\(0 \le \Vert r_1 f_1 + r_2 f_2 \Vert \le \Vert r_1 f_1 \Vert + \Vert r_2 f_2 \Vert = \vert r_1 \vert \Vert f_1 \Vert + \vert r_2 \vert \Vert f_2 \Vert = 0\), so, \(\Vert r_1 f_1 + r_2 f_2 \Vert = 0\), so, \(r_1 f_1 + r_2 f_2 \in \{f \in \mathcal{L}^p (M, A, \mu, F) \vert \Vert f \Vert = 0\}\).
So, \(\{f \in \mathcal{L}^p (M, A, \mu, F) \vert \Vert f \Vert = 0\}\) is a vectors subspace, 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.
\(\{f \in \mathcal{L}^\infty (M, A, \mu, F) \vert \Vert f \Vert = 0\}\) is indeed a vectors subspace, likewise.
Let us see that \(\Vert [f] \Vert\) is well-defined.
Let \([f] = [f'] \in L^p (M, A, \mu, F) \text{ or } L^\infty (M, A, \mu, F)\) be any.
\(\Vert f' \Vert = \Vert f' - f + f \Vert \le \Vert f' - f \Vert + \Vert f \Vert\), because it is a seminorm, but \(\Vert f' - f \Vert = 0\), so, \(\Vert f' \Vert \le \Vert f \Vert\).
Likewise, \(\Vert f \Vert \le \Vert f' \Vert\).
So, \(\Vert f' \Vert = \Vert f \Vert\).
So, the norm is well-defined.
Let us see that \(\Vert [f] \Vert\) is indeed a norm.
Let \(\forall [f_1], [f_2] \in L^p (M, A, \mu, F) \text{ or } L^\infty (M, A, \mu, F)\) and \(r \in F\) be any.
1) (\(0 \le \Vert [f_1] \Vert\)) \(\land\) (\((0 = \Vert [f_1] \Vert) \iff ([f_1] = 0)\)): \(0 \le \Vert f_1 \Vert = \Vert [f_1] \Vert\); if \(0 = \Vert [f_1] \Vert\), \(0 = \Vert f_1 \Vert\), which means that \([f_1] = 0\); if \([f_1] = 0\), \([f_1] = [0]\), and \(\Vert [f_1] \Vert = \Vert 0 \Vert = 0\).
2) \(\Vert r [f_1] \Vert = \vert r \vert \Vert [f_1] \Vert\): \(\Vert r [f_1] \Vert = \Vert [r f_1] \Vert = \Vert r f_1 \Vert = \vert r \vert \Vert f_1 \Vert = \vert r \vert \Vert [f_1] \Vert\).
3) \(\Vert [f_1] + [f_2] \Vert \le \Vert [f_1] \Vert + \Vert [f_2] \Vert\): \(\Vert [f_1] + [f_2] \Vert = \Vert [f_1 + f_2] \Vert = \Vert f_1 + f_2 \Vert \le \Vert f_1 \Vert + \Vert f_2 \Vert = \Vert [f_1] \Vert + \Vert [f_2] \Vert\).
