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I've recently been getting involved with certain mathematical details that I've had to worry about previously. When we define a probability space we say we have a triple $(\Omega, \mathcal F, P)$ where $\Omega$ is our sample space , $\mathcal F$ is a $\sigma-$ Algebra and $P$ is a probability measure. Or when we say a normed space is a pair $(V,||.||)$ where $V$ is our vector space and $||.||$ is the norm function. A similar question to mine is found here but it tackles a different aspect.

Question is what do we really mean by putting these as a pair or a triple? What's the significance? For example never include the basic operations (+ $-$ $\div$ $\times$) in our tuple why do we have to include the norm function $||.||$ ?

AcademicalResearcher
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    Saying "$(\Omega, \mathcal{F}, P)$" is a probability space" is shorter and less tedious than saying "$\mathcal{F}$ is a $\sigma$-algebra on $\Omega$ and $P$ is a probability measure on $\mathcal{F}$. You immediately can identify how the different objects relate because you can read it off the triplet. – J. De Ro Sep 09 '20 at 14:49
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    It is supposed to emphasize that one component alone does not dictate the entire structure: they must all come bundled together. A vector space $V$ could have multiple norms, and changing the norm but keeping the space could have different behavior. You need to study them together. – Randall Sep 09 '20 at 15:06
  • @Randall But do we have to include in our tuple all the operations that can act on elements in there ? Say we want to add dot product (like in Hilbert Spaces) do we have to introduce the triple $(V,||.||, <.,.>)$ where we have a vector space $V$ , a norm and the inner product? The abstraction of why we have to put in there all these things is quite not clear to me yet. Why not also include the basic operations in the tuple .. after all we do need those to act on our vectors in $V$? – AcademicalResearcher Sep 09 '20 at 15:11
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    @AcademicalResearcher it depends on the goals of your writing. – Randall Sep 09 '20 at 15:22
  • Can you please explain a little @Randall (: and thank you – AcademicalResearcher Sep 09 '20 at 15:23
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    Usually, one says "let $V$ be a vector space", and inside this $V$ we also consider the whole structure, $V=(!V,+,-,\cdot,\text{etc.})$, where $!V$ is the underlying set. Then when we understand all these vector spaces, we take the tuple as a whole. Adding only a scalar product is done by saying "... let $V$ be a vector space with a scalar product $\langle\dots\ ,\ \dots\rangle$. Then a Hilbert space $H=(V,\langle\ \ ,\ \ \rangle)$ is this structure witht the properties... – dan_fulea Sep 09 '20 at 16:03
  • @dan_fulea I like your answer. So is the following correct .. By default the vector space V includes the basic operations and when we need to introduce a non basic operation such as inner product or norm we have to put them into a tuple. So lets create a tuple (quadruple in this case) for example $(V,F,<.,.>, ||.||)$ V is a vector space F is a scalar and we have inner product and norm and call it Bla-Space. Can we read this as the BlaSpace can use the basic operations , inner product and norm and it can utilize scalars and vectors from V and all those are needed to fully describe the BlaSpace? – AcademicalResearcher Sep 09 '20 at 16:10
  • I would best put the story in a more categorial context, to have a better order. A category is described here... https://en.wikipedia.org/wiki/Category_(mathematics) Then we start with the category of sets. Then we "add structure" to a set, get a "specialized" category of groups. There is a forgetful functor from the groups to the sets, associating to a group $G$ its underlying set $!G=S$. A posteriori. When defining a group object, we say it is a tuple $(S,\cdot)$ with some axioms. (The axioms should determine $0$, $-$, etc. - if not, feel free to add them to the tuple.) In this manner, we... – dan_fulea Sep 09 '20 at 16:16
  • ... construct one by one the categories of sets, groups, abelian groups, rings, fields, vector spaces over a fixed field $F$, normed vector spaces over some field with a norm, etc. - and at each step we add to the "poor" object of an already understood category some structure. E.g. if we know normed spaces, with objects written $V=(!V,|\ |)$, then a space with an inner product is a $W=(V,\langle\ ,\ \rangle)$, where the norm of $V$ comes from the specified inner product. And all that underlying structure is considered as part of $V$. This makes writing things simpler. The forgetful functor.. – dan_fulea Sep 09 '20 at 16:21
  • ... would be then $!W=V$. – dan_fulea Sep 09 '20 at 16:21

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