# tvs - Chapter 5 Topological Vector Spaces In this chapter V...

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Chapter 5 Topological Vector Spaces In this chapter V is a real or complex vector space. 5.1 Topological Vector Spaces A complex vector space V equipped with a topology is a broad-sense topo- logical vector space if the mappings V × V V : ( x,y ) 7→ x + y C × V V : ( λ,x ) 7→ λx are continuous. Observe that then, for each x V , the translation map τ x : V V : y 7→ y + x is continuous. Since τ - 1 x = τ - x , it follows that τ x is a homeomorphism. The simple but important consequence of this is that V “looks the same everywhere”, i.e. if a,b V then there is a homeomorphism , speciﬁcally τ b - a : V V , which maps a to b . In particular, every neighborhood of x V is a translate of a neighborhood of 0, i.e. of the form x + U for some neighborhood U of 0. For this reason, we shall prove most of our results in a neighborhood of 0. By a topological vector space we shall mean a broad-sense topological vector space which is Hausdorﬀ , i.e. distinct points of disjoint neighborhoods. Lemma 1 Let V be a broad-sense topological vector space, and W an open set with 0 W . Then there is an open set U with 0 U , U = - U , and U + U W 1

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2 CHAPTER 5. TOPOLOGICAL VECTOR SPACES Proof . Since V × V V : ( x,y ) 7→ x + y is continuous at 0, and W is a neighborhood of 0, there is a neighborhood U 1 of 0 such that U 1 + U 1 W To get symmetry take U = U 1 ( - U 1 ) Note that continuity of multiplication by scalars implies that x 7→ - x is a homeomorphism and so - U 1 is open when U 1 is open. QED Here is a simple but useful observation: if A is any subset of the broad- sense topological vector space V and U an open subset of V then the set A + U = { a + x : a A,x U } is open. The reason is that
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tvs - Chapter 5 Topological Vector Spaces In this chapter V...

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