Could someone me to read the following line

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Discussion Overview

The discussion revolves around the interpretation of the term "map" in the context of inner products in vector spaces. Participants are exploring the definition and implications of this term within mathematical structures, specifically relating to vector spaces and fields.

Discussion Character

  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant explains that the inner product ##\langle \cdot, \cdot \rangle## is a map from the Cartesian product of a vector space ##V## with itself to a field ##F##.
  • Another participant seeks clarification on the meaning of "map" in this context, referencing a general definition of the term.
  • A later reply confirms that in mathematics, "map" and "function" are synonymous, suggesting a shared understanding of the terminology.

Areas of Agreement / Disagreement

Participants generally agree on the definition of "map" as it relates to functions in mathematics, but there is an ongoing request for further clarification on its specific application to the inner product.

Contextual Notes

The discussion does not resolve the nuances of how "map" is applied in this specific mathematical context, leaving some assumptions and definitions potentially unaddressed.

EnglsihLearner
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Could someone please help me to read the following line(in case of inner product)?
94842b9bc828789f99f1ff660295844b.png
 
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The statement indicates that ##\langle \cdot, \cdot \rangle## is a map from ##V \times V## to ##F##. This means that to each ordered pair ##(u,v)## with ##u,v \in V##, it assigns an element ##\langle u,v \rangle## in ##F##.

Presumably ##F## is a field, and ##V## is a vector space over ##F##.
 
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jbunniii said:
The statement indicates that ##\langle \cdot, \cdot \rangle## is a map from ##V \times V## to ##F##.
Yes, I understand now. Could you please explain what does "map" mean here?As I know map means the following-

https://www.google.com.bd/search?bi....0..0.0...0...1c.1.24.serp..3.0.0.hrQDtAr7Tpw

If it is same here then I would request you to explain again the following part-

that ##\langle \cdot, \cdot \rangle## is a map from ##V \times V## to ##F##
 
EnglsihLearner said:
Yes, I understand now. Could you please explain what does "map" mean here?
In mathematics, "map" and "function" mean the same thing.
 

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