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-40,7 +40,36 @@ \[ d_4(3, 2) = |3 - 2\cdot 2| = 1 \neq 4 = |2 - 2\cdot 3| = d_4(2,3) .\] - \item + \item $d_5(x,y) = \frac{|x-y|}{1 + |x-y|}$ ist eine Metrik, denn $\forall x, y, z \in \R$ gilt: + \begin{enumerate}[(M1)] + \item $d_5(x, y) = \frac{|x-y|}{1 + |x-y|} \ge 0$ und + \[ + d_5(x, y) = 0 \iff \frac{|x-y|}{1 + |x-y|} = 0 + \iff |x - y| = 0 \iff x = y + .\] + \item $d_5(x, y) = \frac{|x-y|}{1 + |x-y|} = \frac{|y-x|}{1 + |y-x|} = d_5(y, x)$. + \item Sei $d := \max \{ |x-y|, |x-z|, |y-z| \} $. Falls $d = |x-z|$, dann ist + \[ + \frac{|x-z|}{1 + |x-z|} = \frac{|x-z|}{1+d} + \le \frac{|x-y|}{1 + d} + \frac{|y-z|}{1+d} + \le \frac{|x-y|}{1 + \underbrace{|x - y|}_{\le d}} + \frac{|y-z|}{1 + + \underbrace{|y-z|}_{\le d}} + .\] + Falls $d \neq |x-z|$. Dann sei O.E. $d = |x-y|$. Dann gilt + \begin{alignat*}{2} + &\quad&|x-z| &\le |x-y| \\ + \iff& &|x-z| + |x-y|\cdot |x-z| &\le |x-y| + |x-y| \cdot |x-z| \\ + \iff& &|x - z| (1 + |x-y|) &\le |x-y| (1 + |x-z|) \\ + \iff& &\frac{|x-z|}{1 + |x-z|} &\le \frac{|x-y|}{1 + |x-y|} + \intertext{Also insbesondere} + \implies& &\frac{|x-z|}{1 + |x-z|} &\le \frac{|x-y|}{1+|x-y|} + + \underbrace{\frac{|y-z|}{1 + |y-z|}}_{\ge 0} + .\end{alignat*} + Insgesamt folgt + \[ + d_5(x,z) \le d_5(x,y) + d_5(y,z) + .\] + \end{enumerate} \end{enumerate} \end{aufgabe} @@ -72,23 +101,53 @@ \begin{aufgabe} Seien $p, q \in \R$ mit $p, q > 1$, $\frac{1}{p} + \frac{1}{q} = 1$ \begin{enumerate}[(a)] - \item Definiere - \begin{align*} - f(t) &:= (a^{p})^{t} \cdot (b^{q})^{1 - t} - \intertext{Es folgt} - f''(t) &= \underbrace{(p \ln a - q \ln b)^2}_{\ge 0} \cdot \underbrace{f(t)}_{\ge 0} - .\end{align*} - Also ist $f(t)$ konvex. - Es ist mit $\frac{1}{p} + \frac{1}{q} = 1 \implies \frac{1}{q} = 1 - \frac{1}{p}$. - Mit $\lambda = \frac{1}{p}, x = 1$ und $y = 0$ folgt damit + \item Beh.: Für $a, b \ge 0$ gilt + \[ + ab \le \frac{a^{p}}{p} + \frac{b^{q}}{q} + .\] + \begin{proof} + Definiere + \begin{align*} + f(t) &:= (a^{p})^{t} \cdot (b^{q})^{1 - t} = e^{(p \ln a - q \ln b) t} + \intertext{Es folgt} + f''(t) &= \underbrace{(p \ln a - q \ln b)^2}_{\ge 0} \cdot \underbrace{f(t)}_{\ge 0} + .\end{align*} + Also ist $f(t)$ konvex. + Es ist mit $\frac{1}{p} + \frac{1}{q} = 1 \implies \frac{1}{q} = 1 - \frac{1}{p}$. + Mit $\lambda = \frac{1}{p}, x = 1$ und $y = 0$ folgt damit + \[ + ab = + a^{\frac{1}{p} \cdot p}b^{q \cdot \left( 1-\frac{1}{p}\right) } + = f\left(\frac{1}{p}\right) + \le \frac{1}{p} f(1) + \left( 1 - \frac{1}{p} \right)f(0) + = \frac{a^{p}}{p} + \frac{b^{q}}{q} + .\] + \end{proof} + \item Beh.: Für $a_1, b_1, \ldots, a_n, b_n \in \R$ gilt \[ - ab = - a^{\frac{1}{p} \cdot p}b^{q \cdot \left( 1-\frac{1}{p}\right) } - = f\left(\frac{1}{p}\right) - \le \frac{1}{p} f(1) + \left( 1 - \frac{1}{p} \right)f(0) - = \frac{a^{p}}{p} + \frac{b^{q}}{q} + \sum_{i=1}^{n} |a_i b_i| \le \left( \sum_{i=1}^{n} |a_i|^{p} \right)^{\frac{1}{p}} + \cdot \left( \sum_{i=1}^{n} |b_i|^{q} \right)^{\frac{1}{q}} .\] - \item + \begin{proof} + Es sei $\Vert a \Vert_p := \left(\sum_{i=1}^{n} |a_i|^{p}\right)^{\frac{1}{p}}$ + und $\Vert b \Vert_q := \left( \sum_{i=1}^{n} |b_i|^{q} \right)^{\frac{1}{q}} $. Dann folgt + \begin{align*} + \frac{1}{\Vert a \Vert_p \cdot \Vert b \Vert_q} + \sum_{i=1}^{n} |a_i b_i| + &= \quad\sum_{i=1}^{n} \left|\frac{a_i}{\Vert a \Vert_p}\right| + \left| \frac{b_i}{\Vert b \Vert_q} \right|\\ + &\stackrel{\text{Young}}{\le} \quad + \sum_{i=1}^{n} \frac{|a_i|^{p}}{p \Vert a \Vert_p^{p}} + + \sum_{i=1}^{n} \frac{|b_i|^{q}}{q \Vert b \Vert_q^{q}} \\ + &= \frac{1}{p} \frac{1}{\sum_{i=1}^{n} |a_i|^{p}} + \sum_{i=1}^{n} |a_i|^{p} + + \frac{1}{q} \frac{1}{\sum_{i=1}^{n} |b_i|^{q}} + \sum_{i=1}^{n} |b_i|^{q} \\ + &= \frac{1}{p} + \frac{1}{q} \\ + &= 1 \\ + \implies \sum_{i=1}^{n} |a_i b_i| &\le \Vert a \Vert_p \cdot \Vert b \Vert_q + .\end{align*} + \end{proof} \end{enumerate} \end{aufgabe}