TY - JOUR AU - Li, Chi-Kwong AU - Tsing, Nam-Kiu PY - 1990/03/01 Y2 - 2024/03/29 TI - NORMS ON CARTESIAN PRODUCT OF LINEAR SPACES JF - Tamkang Journal of Mathematics JA - Tamkang J. Math. VL - 21 IS - 1 SE - Papers DO - 10.5556/j.tkjm.21.1990.4692 UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/4692 SP - 35-39 AB - <div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">Let $</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">X_i, </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">(i=1, \cdots, </span><span style="font-size: 10.000000pt; font-family: 'Arial'; font-style: italic;">n)$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">be real or complex linear spaces, each equipped with a norm $||\cdot||_i$</span><span style="font-size: 13.000000pt; font-family: 'ArialMT';">. </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">Standard ways of constructing norms </span><span style="font-size: 13.000000pt; font-family: 'ArialMT';">$||\cdot||$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">on the Cartestian product $</span><span style="font-size: 9.000000pt; font-family: 'Arial'; font-style: italic;">X </span><span style="font-size: 15.000000pt; font-family: 'TimesNewRomanPSMT';">=</span><span style="font-size: 9.000000pt; font-family: 'Arial'; font-style: italic;">X_1</span><span style="font-size: 7.000000pt; font-family: 'ArialMT';"> \times \cdots \times </span><span style="font-size: 9.000000pt; font-family: 'Arial'; font-style: italic;">X_</span><span style="font-size: 7.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">n$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">are to define</span></p><p><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">\[ ||(x_1, \cdots, x_n)||=\phi(||x_1||_1, \cdots, ||x_n||_n)\]</span></p><div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">via some functions $\phi$</span> <span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">on $\mathbb{</span><span style="font-size: 9.000000pt; font-family: 'ArialMT';">R}^n$. </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">Common examples of $\phi$</span> <span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">in standard texbooks are </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">norms on $\mathbb{<span style="font-size: 9.000000pt; font-family: 'ArialMT';">R}^n$</span></span><span style="font-size: 9.000000pt; font-family: 'ArialMT';">. </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">This may mislead peoples to think that any norm </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">$\phi$ on </span><span style="font-size: 9.000000pt; font-family: 'ArialMT';"><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">$\mathbb{</span>R}^n$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">can </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">induce a norm on the product space $</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">X$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">in the above way. </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">In </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">this note we show that </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">this is actually false and characterize the functions </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;"><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">$\phi$</span> </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">that can give rise to norms </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">on $</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">X$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">in the above manner. It turns out that a necessary and sufficient condition on $\phi$</span> <span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">is :</span></p><div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">for </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">any $</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">a_1, \cdots, a_n, </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">b_1, \cdots, b_n\ge 0$</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">,<br /></span></p><div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">(I) <span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">$\phi</span></span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">(<span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">a_1, \cdots, a_n</span>)</span><span style="font-size: 12.000000pt; font-family: 'ArialMT';">&gt;</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">0$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">if $(<span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">a_1, \cdots, a_n</span></span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">)</span><span style="font-size: 9.000000pt; font-family: 'Arial'; font-style: italic;"> eq </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">(0, \cdots,0)$; </span></p><p><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">(II) <span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">$\phi</span>(\alpha(a_1, \cdots, a_n))= \alpha \phi</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">(<span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">a_1, \cdots, a_n</span></span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">)$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">if $\alpha\ge 0$</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">; </span></p><p><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">(III) <span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">$\phi</span>(c_1, \cdots, c_n)\le \phi</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">(</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">a_1, \cdots, a_n)+ \phi</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">(b<span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">_1, \cdots, b_n</span>)$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">if $</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">(c_1, \cdots, c_n)= (a_1, \cdots, a_n</span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">)+ (b<span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">_1, \cdots, b_n</span>)$;<br /></span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">(IV) <span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">$\phi</span></span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">(<span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">a_1, \cdots, a_n</span>) \le \phi(b<span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">_1, \cdots, b_n</span>)$ </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">if $a_i \le </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">b_i$ </span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPSMT';">for </span><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">all $</span><span style="font-size: 10.000000pt; font-family: 'TimesNewRomanPS'; font-style: italic;">i$. </span></p><div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 9.000000pt; font-family: 'TimesNewRomanPSMT';">Several interesting consequences of the result are discussed. </span></p></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div> ER -