By Lynn Arthur Steen, J. Arthur Seebach Jr.

Over a hundred and forty examples, preceded by means of a succinct exposition of common topology and easy terminology. every one instance handled as a complete. Over 25 Venn diagrams and charts summarize houses of the examples, whereas discussions of common tools of building and alter provide readers perception into developing counterexamples. comprises difficulties and routines, correlated with examples. Bibliography. 1978 variation.

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**Extra info for Counterexamples in Topology (Dover Books on Mathematics)**

**Example text**

E. the element f , does not belong to the range R(L ) of an operator L . 1) 1 Note that studying of a closable operator L : E → F can be reduced (at least theoretically) to studying of a linear continuous operator L1 defined on the same set D(L ), but with respect to another norm. Indeed, introducing in D(L ) a graph norm u Γ = u E + Lu F, with respect to which the linear set D(L ) is Banach, we have that the operator L1 : D(L ) → F is linear and continuous (L1 u = L u, u ∈ D(L ) = D(L1 )). 10 2 The Simplest Schemes ...

This isometry defines the completion L¯ of the operator L . Thus, the operator L¯ sets an isometry between E¯ and F. This implies the above-mentioned properties of L¯ . The foregoing implies the following theorem. 1. 1. If f ∈ R(L ), then a strong generalized solution u¯ turns into a classic solution. It is also clear that the classic solution is strong, and it is classic if u¯ ∈ D(L ). ¯ Since D(L ) is a dense Let us clarify the relations between the spaces E and E. linear subset of E (of course, in the sense of the norm of the space E), then the set E may be obtained by completing D(L ) with respect to the norm u E .

Indeed, u M¯ |L ∗ ϕ (u)| |ϕ (L u)| = sup = Lu ϕ F∗ ϕ F∗ ϕ ∈F ∗ L ∗ ϕ ∈M = sup F. This equality holds because, by virtue of Hahn–Banach Theorem, for any element L u ∈ F there exists such a functional ϕ ∈ F ∗ with unit norm that ϕ (L u) = L u F . In addition, we have the following lemma. 2. If M ⊂ R(L ∗ ) and MF is a total subset of the space F ∗ , then the space ¯ E¯L is embedded into the space M. Proof. The totality of the set MF and the injectivity of the operator L imply the totality of the set M.