Structure Theorem for Functionals in the Space 𝔖′𝜔1,𝜔2 - Hamed M. Obiedat; Wasfi A. Shatanawi; Mohd M. Yasein (International Journal of Mathematics and Mathematical Sciences)
We introduce the space 𝔖𝜔1,𝜔2 of all 𝐶∞ functions 𝜙 such that 𝑠𝑢𝑝|𝛼|≤𝑚‖𝑒𝑘𝜔1𝜕𝛼𝜙‖∞ and 𝑠𝑢𝑝|𝛼|≤𝑚‖𝑒𝑘𝜔2𝜕𝛼0𝑥0005𝑒𝜙‖∞ are finite for all 𝑘∈ℕ0, 𝛼∈ℕ𝑛0, where 𝜔1 and 𝜔2 are two weights satisfying the classical Beurling conditions. Moreover, we give a topological characterization of the space 𝔖𝜔1,𝜔2 without conditions on the derivatives. For functionals in the dual space 𝔖𝜔1,𝜔2, we prove a structure theorem by using the classical Riesz representation thoerem.