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Combined display of all available logs of ProofWiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).
- 05:58, 16 January 2022 Addem talk contribs created page User:Addem/riesz fischer (Created page with "== Theorem == Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $p \in \R$, $p \ge 1$. The Lebesgue $p$-space $\map {\LL^p} \mu$, endowed with the $p$-norm, is a complete metric space. == Proof == To show that $\map {\LL^p} \mu$ is complete we consider an arbitrary norm Cauchy sequence $\{f_n\}$ taken from $\map {\LL^p} \mu$, and demonstrate th...")
- 18:00, 14 January 2022 Addem talk contribs created page User:Addem/Holder/Integral (Created page with "== Theorem == Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $p, q \in \R$ such that $\dfrac 1 p + \dfrac 1 q = 1$. Let $f \in \map {\LL^p} \mu, f: X \to \R$, and $g \in \map {\LL^q} \mu, g: X \to \R$, where $\LL$ denotes Lebesgue space. Then their pointwise product $f g$ is $\mu$-integrable, that is: :$f g \in \map...")
- 17:48, 14 January 2022 Addem talk contribs created page User:Addem/Holder/Summation (Created page with "== Theorem == <onlyinclude> Let $p, q \in \R_{>0}$ be strictly positive real numbers such that: :$\dfrac 1 p + \dfrac 1 q = 1$ Let: :$\mathbf x = \sequence {x_n} \in \ell^p$ :$\mathbf y = \sequence {y_n} \in \ell^q$ where $\ell^p$ denotes the $p$-sequence space. Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$. Then $\mathbf x \mathbf y = \sequence {x_n y_n}...")
- 17:38, 14 January 2022 Addem talk contribs created page User:Addem/Holder (Created page with "== Hölder's Inequality == There are two theorems which go by the title of Hölder's Inequality, one dealing with summation and the other with integration with respect to a measure. In both cases it is asserted that, when $1\le p\le \infty$ and $q$ is the exponential conjugate of $p$, then $ \|ab\|_1 \le \|a\|_p\|b\|_q $ In the summation case we interpret $\|\cdot \|_p$ as the P-norm and in the integral case we interpret it as The result for summation can be d...")
- 17:04, 14 January 2022 Addem talk contribs created page Talk:Finite Union of Open Sets in Complex Plane is Open (why not infinite)