Let I ⊆ R be a noncompact interval and let ρ: I → [1,∞) be an increasing and differentiable function called weight. Let Bρ(I) be the space of all functions f : I → R such that |f(x)| ≤ M · ρ(x), for all x ∈ I, where M > 0 is a constant depending on f and ρ, but independent of x. The space Bρ(I) is called weighted space and it is a Banach space endowed with the ρ-norm ||f||ρ = sup_{x∈I} [|f(x)|/ ρ(x)]. Please suggest a reference from which proof of this result can be found.
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