Meaning of 'Space': Vector, Banach & Set Definitions

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Discussion Overview

The discussion revolves around the meaning of 'Space' in mathematical contexts, specifically regarding vector spaces, Banach spaces, and related definitions. Participants explore the conceptual underpinnings and properties that define various types of spaces.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions whether 'space' is merely another term for a set.
  • Another participant suggests that a space is generally a set or collection of mathematical objects with specific properties and operations.
  • A further example is provided regarding Boolean Space, illustrating a set with defined operations and axioms.
  • A light-hearted remark is made about the terminology used in mathematical definitions, specifically referencing Banach spaces.

Areas of Agreement / Disagreement

Participants express differing views on the definition of 'space,' with some emphasizing its properties and others questioning its equivalence to a set. The discussion remains unresolved regarding a unified definition.

Contextual Notes

Participants do not fully explore the implications of their definitions, and there may be missing assumptions regarding the properties that distinguish different types of spaces.

Swapnil
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What is the meaning of 'Space' (in the context of vector spaces, Banach spaces, etc)? Is space just another name for a set?
 
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It depends on the context... But in general, a space is a set (or class or collection of mathematical objects) with some special properties (e.g. it is equipped with certain operations which in turn satisfy certain requirements). In essense, it's the universe you're going to work in, hence the name.
 
So, for example, a Boolean Space [tex]\mathcal{B}[/tex] would be a set of two elements 0 and 1 equipped with two binary operations [tex]\lor[/tex] and [tex]\land[/tex] and an unary operation [tex]\lnot[/tex] such that the usual axioms of associativity, commutativity, distributivity, etc hold. Right?
 
Last edited:
well it sounds better than banach doo hickey.
 

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