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| Paper Title: | Convex Conjugate of A Bounded Linear Functional on L^p- Space |
| Authors Name: | T Srinivasarao |
| Unique Id: | IJSDR2207095 |
| Published In: | Volume 7 Issue 7, July-2022 |
| Abstract: | If Γ:L^p→ L^q is a linear functional having 1/p+1/q=1, for each g∈L^q that realizes the relevant f∈L^p such that Γ(f)= ∫_0^1▒〖f∙g〗 with ‖Γ‖= ‖g‖_q, then there is a convex complement of Γ given by ⋀:L^q→ L^p with the property αΓ+β⋀= 1 for some 0<α<1 and β=1-α. if α=0 or α = 1, then the functionals Γand ⋀ will become singular and so, will not satisfy the contraction principle. Recollect that, if {φ_n } converges to f in L^p- space, then ‖Γ(f)-Γ(φ_n)‖_p≤‖Γ‖ ‖f-φ_n ‖_p and by the properties of the linear functionalΓ, it is known that‖Γ‖=‖g‖_q. Using this, ‖Γ(f)-Γ(φ_n)‖_p≤‖g‖_q ‖f-φ_n ‖_p. since L^p& L^q are linear spaces on the set of real numbers with the symmetric condition 1/p+1/q=1, by interchanging the roles of f and g in the Riesz – representation theorem, for each ⋀:L^q→ L^p and g∈L^q, there corresponds a unique f∈L^p with 1/p+1/q=1 such that ‖⋀(g)-⋀(ψ_n)‖_q≤‖⋀‖ ‖g-ψ_n ‖_q |
| Keywords: | Lp - Space, Convexity of a Linear Operator on an Lp-Space, Norm of a Linear Operator |
| Cite Article: | "Convex Conjugate of A Bounded Linear Functional on L^p- Space", International Journal of Science & Engineering Development Research (www.ijsdr.org), ISSN:2455-2631, Vol.7, Issue 7, page no.631 - 632, July-2022, Available :http://www.ijsdr.org/papers/IJSDR2207095.pdf |
| Downloads: | 000337072 |
| Publication Details: | Published Paper ID: IJSDR2207095 Registration ID:201067 Published In: Volume 7 Issue 7, July-2022 DOI (Digital Object Identifier): Page No: 631 - 632 Publisher: IJSDR | www.ijsdr.org ISSN Number: 2455-2631 |
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