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Familiarization with Semi-normed Spaces

A semi-normed space E is a vector space endowed with a family of semi-norms.

— The set NE indexing the semi-norms is, a priori, arbitrary.

— A normed space is the special case where this family simply consists of a single norm.

— Every locally convex topological vector space can be endowed with a family of semi-norms that generates its topology (Neumann’s theorem).

— We only consider separated spaces, namely in which ||u||_E;v = 0 for every v ∈ NE, then u = 0E.

Working with semi-normed spaces:

— un → u in E means that ||un − u||_E;v → 0 for every v ∈ NE.

— U is bounded in E means that for every v ∈ NE.

— T is continuous from F into E at the point u means that, for every v ∈ NE and ϵ > 0, there exists a finite set M of NF and η > 0 such that implies .

Examples — real-valued function spaces:

— 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”.