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Functional analysis is considered as very important aspects in mathematics, applied sciences and also is very powerful way of examining the behaviour of various mathematical models and it clarifies, regresses and unifies the underlying concepts in mathematics, engineering, economics other applied fields.
Hilbert space is a Banach space because Hilbert space is complete with respect to the norm associated with its inner product where a norm and an inner product are said to be associate if ‖X‖2 = 〈X,X〉 for all X. Moreover the converse is not always true, not every Banach space is a Hilbert space. A necessary and sufficient condition for a Banach space X to be associated to an inner product (which will then necessarily make X into a Hilbert space) is the parallelogram Identity. This implies that Hilbert space is inner product spaces, Banach space are normed spaces and complete metric spaces are metric space.
The Brouwer’s fixed point theorem is one of the most well-known and useful theorem in topology. Since the theorem and its many extensions are powerful tools in showing the existence of solutions of many problems in pure and applied mathematics, many scholars have been studying its further extensions and applications.
Brouwer’s fixed point theorem has always been a major theoretical tool which can be applied in differential equations, topology, economics, games theory, dynamics and functional analysis. Moreover, more or recently, the usefulness of the concept for the applications increased enormously by the development of accurate and efficient techniques for computing fixed points. |
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