Discussion on Some Common Properties of Hilbert Space and Topological Space

Abstract

Theory of elliptic equations and undergone considerable progress that has created the concept of the topological character and eventually has created interesting connection with the analysis and Hilbert Space. Connection of Hilbert space and Topological space has been a matter of curiosity for many. Although there is no much work done on this topic, the answer to this question is Hilbert space is a part of the Topological space only when there is a need of functional analysis. In other words topology is induced to the Hilbert space but in real they are not a part of each other. In algebra concept, topology is involved in the Hilbert spaces to support the idea of metric space. Topological space consists of abstract sets of points that includes specific collection open sets of subsets that need to satisfy the axioms. Hausdorff space is one of the types of topological space .Lot of properties are satisfied by the Hausdorff space which are not satisfied by other way. Unlike the Hilbert space, the topological space is not highly complicated and forms the basis of the functional analysis. Two things, a topological space and one special type of vector are present at once in the Hilbert space. Thus, in Hilbert space more topological structure is given by the topological space while the special type of vector would help in giving some algebraic space.

Keywords

• Hilbert space
• Topological space
• Hausdorff space
• Inner product space
• Γ- Hilbert space

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