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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
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diff --git a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md new file mode 100644 index 0000000..023c253 --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md @@ -0,0 +1,36 @@ +--- +title: The L<sup>p</sup> Spaces +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 4 +--- + +# {{ page.title }} + +{% definition %} +Let $(X,\mathcal{A},\mu)$ be a measure space and let $p \in [1,\infty)$. +We write $\mathscr{L}^p(X,\mathcal{A},\mu)$ for the set of all +measurable functions $f : X \to \KK$ such that $\abs{f}^p$ is integrable. +For such $f$ we write + +$$ +\norm{f}_p = {\bigg\lparen\int_X \abs{f}^p \, d\mu\bigg\rparen}^{\!1/p}. +$$ +{% enddefinition %} + +{% proposition %} +Endowed with pointwise addition and scalar multiplication +$\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space. +{% endproposition %} + +{% proposition %} +$\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$. +{% endproposition %} + +{% theorem * Young Inequality %} +Consider $p,q > 1$ such that $1/p + 1/q = 1$. Then + +$$ +a \cdot b \le \frac{a^p}{p} + \frac{b^q}{q} \qquad \forall a,b \ge 0. +$$ +{% endtheorem %} |