TY - JOUR AU - Akweittey, Emmanuel AU - Obed Fosu, Gabriel AU - Mubarrack, Ahmed PY - 2016/06/25 Y2 - 2024/03/29 TI - On the Merits of Two Different Topologies on the Dual of a Hilbert Space JF - Asian Journal of Applied Sciences JA - AJAS VL - 4 IS - 3 SE - Articles DO - UR - https://ajouronline.com/index.php/AJAS/article/view/3908 SP - AB - <span style="font-family: CMBXTI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><strong><em>Two topologies of a Hilbert space </em></strong><span style="font-family: CMMI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><em>l </em><span style="font-family: CMR7; font-size: 6pt; color: #000000; font-style: normal; font-variant: normal;">2 <span style="font-family: CMBXTI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><strong><em>is considered in this work. To wit normed</em></strong><br /><span style="font-family: CMBXTI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><strong><em>vector space, which is its natural topology and weak topology. Among other properties we showed</em></strong><span style="font-family: CMBXTI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><strong><em> that the closed unit ball </em></strong><span style="font-family: CMMI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><em>S </em><span style="font-family: CMBXTI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><strong><em>is not compact when </em></strong><span style="font-family: CMMI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><em>l </em><span style="font-family: CMR7; font-size: 6pt; color: #000000; font-style: normal; font-variant: normal;">2<span style="font-family: CMSY7; font-size: 6pt; color: #000000; font-style: normal; font-variant: normal;"><em>∗ </em><span style="font-family: CMBXTI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><strong><em>is given its topology as a Banach space. On the</em></strong><span style="font-family: CMBXTI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><strong><em> other hand </em></strong><span style="font-family: CMMI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><em>S </em><span style="font-family: CMBXTI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><strong><em>is compact when </em></strong><span style="font-family: CMMI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><em>l </em><span style="font-family: CMR7; font-size: 6pt; color: #000000; font-style: normal; font-variant: normal;">2<span style="font-family: CMSY7; font-size: 6pt; color: #000000; font-style: normal; font-variant: normal;"><em>∗ </em><span style="font-family: CMBXTI10; font-size: 9pt; color: #000000; font-style: normal; font-variant: normal;"><strong><em>is given its weak topology.</em></strong></span></span></span><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;" /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> ER -