• DR RAJNI

The present article explains proof of The Hahn‐Banach theorem which is one of the most important theorems in functional analysis . In Hahn‐Banach theorem we have shown that all of the bounded linear functionals on a Hilbert space are just the scalar products, you might begin to wonder whether Banach spaces which are not Hilbert spaces have any non,zero bounded linear functionals at all. (In fact, one might be tempted to reason as follows: if the only nontrivial bounded linear functionals on a Banach space that has a scalar product are precisely the scalar products, then Banach spaces which do not have scalar products should have none.) we consider the simplest case possible.

bounded , functional, sublinear , vector space , ambitious , norm.

"THE HAHN BANACH THEOREM IN FUNCTIONAL ANALYSIS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.10, Issue 7, page no.j770-j775, July-2023, Available :http://www.jetir.org/papers/JETIR2307998.pdf

"THE HAHN BANACH THEOREM IN FUNCTIONAL ANALYSIS", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.10, Issue 7, page no. ppj770-j775, July-2023, Available at : http://www.jetir.org/papers/JETIR2307998.pdf