Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 360448 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.3700 1210552 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.3700 1210552 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.5014 4196640 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.5029 4224240 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.5037 4241952 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.5039 4272472 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.5050 4295192 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.5050 4295192 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.5699 4985024 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.5737 5177176 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.5742 5198104 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.5866 5418408 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.5867 5418664 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182 Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; JCommentsACL has a deprecated constructor in /var/www/astarmathsandphysics/components/com_jcomments/classes/acl.php on line 17 Call Stack: 0.0001 360448 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.3700 1210552 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.3700 1210552 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.5014 4196640 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.5029 4224240 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.5037 4241952 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.5039 4272472 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.5050 4295192 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.5050 4295192 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.5699 4985024 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.5737 5177176 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.5742 5198104 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.9019 13845472 13. JEventDispatcher->trigger() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:199 0.9021 13845872 14. plgContentJComments->update() /var/www/astarmathsandphysics/libraries/joomla/event/dispatcher.php:160 0.9021 13845872 15. plgContentJComments->onContentAfterDisplay() /var/www/astarmathsandphysics/libraries/joomla/event/event.php:70 0.9023 13854120 16. plgContentJComments->onAfterDisplayContent() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:339 0.9026 13855856 17. JComments::show() /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php:282 0.9033 13888904 18. JCommentsFactory::getACL() /var/www/astarmathsandphysics/components/com_jcomments/jcomments.php:188 0.9033 13889304 19. spl_autoload_call() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274 0.9034 13889384 20. JLoader::load() /var/www/astarmathsandphysics/components/com_jcomments/classes/factory.php:274

Banach Spaces

A Banach space is a complete normed vector space, a vector spaceover a subfield of the complex numberswith a normsuch that every Cauchy sequence inhas a limit in As for general vector spaces, a Banach space overis called a real Banach space, and a Banach space overis called a complex Banach space.

The familiar Euclidean spaceswith norm ofdefined byare Banach spaces. Every finite-dimensional vector space inorbecomes a Banach space on defining a norm, since all norms are equivalent on a finite-dimensionalofvector space.

The set of all continuous functionsdefined on a closed intervalbecomes a Banach space if an appropriate norm is defined in it e.g.known as the supremum norm. This is a well-defined norm since continuous functions defined on a closed interval are bounded.

Sinceis a continuous function on a closed interval, it is bounded and the supremum is attained onso the supremum is the maximum value ofon

The space is complete under this norm, and the resulting Banach space is denoted byThis example can be generalized to the spaceof all continuous functions whereis a compact space, or to the space of all bounded continuous functionswhereis any topological space, or indeed to the spaceof all bounded functionswhereis any set. In all these examples, we can multiply functions and stay in the same space: all these examples are in fact Banach algebras.

Ifis a real number, the space of all infinite sequencesof elements in such that the infinite seriesis finite. Theroot of this sum is the-norm of the sequence. The space, together with this norm, is a Banach space; it is denoted by

The Banach spaceconsists of all bounded sequences of elements inand the norm of such a sequence may be defined as the supremum of the magnitude of the sequence.

Ifare Banach spaces, then we can form their direct sumwhich has a topological vector space structure but no canonical norm, but is a Banach space for several equivalent norms, for example

This construction can be generalized to define-direct sums of arbitrarily many Banach spaces.

If is a closed linear subspace of the Banach spacethen the quotient spaceis again a Banach space.

Every inner product gives rise to an associated norm. The inner product space is called a Hilbert space if its associated norm is complete. Thus every Hilbert space is a Banach space by definition.

Add comment

Security code
Refresh