Continuous Functions. Jacques Simon

      — The space C_b(Ω) of continuous and bounded functions is endowed with the norm

      — C(Ω) is endowed with the semi-norms indexed by the compact sets K ⊂ Ω.

      — Lp(Ω) is endowed with the norm .

      — is endowed with the semi-norms indexed by the bounded open sets ω such that ϖ ⊂ Ω.

       Examples — abstract-valued function spaces:

      — C_b(Ω; E) is endowed with the semi-norms indexed by v ∈ NE

      — C(Ω; E) is endowed with the semi-norms indexed by the compact sets K ⊂ Ω and v ∈ NE

      — L^p(Ω; E) is endowed with the semi-norms indexed by v ∈ NE.

       Examples — weak space, dual space:

      — E-weak is endowed with the semi-norms indexed by e′ ∈ E′.

      — E′ is endowed with the semi-norms indexed by the bounded sets B of E.

      — E′-weak is endowed with the semi-norms indexed by e″ ∈ E″.

      — E′-_*weak is endowed with the semi-norms indexed by e ∈ E.

Neumann spaces and others:

      — A sequentially complete space is a space in which every Cauchy sequence converges.

      — A Neumann space is a sequentially complete separated semi-normed space.

      — A Fréchet space is a sequentially complete metrizable semi-normed space.

      — A Banach space is a sequentially complete normed space.

       Advantages of using semi-norms rather than topology:

      — Semi-norms allow the definition of Lp(Ω; E) (by raising the semi-norms of E to the power p).

      — They allow the definition of the differentiability of a mapping from a semi-normed space into another (by comparing the semi-norms of an increase in the variable to the semi-norms of the increase in the value).

      — They are easy to manipulate: working with them is just like working with normed spaces, the main difference being that there are several semi-norms or norms instead of a single norm.

      — Some definitions are simpler, for example that of a bounded set for any semi-norm || ||E;v of E” would be expressed, in terms of topology, in the more abstract form “for any open set V containing 0_E, there is t > 0 such that tU ⊂ V”.

      Notations

      SPACES OF FUNCTIONS

	space of uniformly continuous functions with bounded support
	space of continuous functions
	space of bounded continuous functions
	space of continuous functions with support included in the compact set K ⊂ Ω
	space of gradients of continuous functions
	set of positive continuous real functions
	space of m times continuously differentiable functions, and the case m = ∞
	id. with bounded derivatives, and the case m = ∞
	id. with support included in the compact set K ⊂ Ω, and the case m = ∞
	space defined on the closure of a bounded open set
	set of functions in taking values in the set U
C(Ω; E)	space of uniformly continuous functions
Cb(Ω; E)	space of bounded uniformly continuous functions
CD (Ω; E)