Trouble understanding the definition of a subspace
$begingroup$
This is a very simple, probably silly, question, but I'm having a diffuclty in unerstanding the terminology used in definition of a subspace.
From Wikipedia:
If $V$ is a vector space over a field $K$ and if $W$ is a subset of $V$, then
$W$ is a subspace of $V$ if under the operations of $V$, $W$ is a vector space
over $K$.
To me, the first sentence seems to be using two different (though related) terms interchangeably: "space" and a "subset".
In other words, why is it correct to say that "$W$ is subset of $V$", when $V$ is not merely a (super)set but a space? Am I being too pedantic over the phrasing?
linear-algebra vector-spaces
asked Dec 14 '18 at 13:29
HeyJudeHeyJude
1687
$endgroup$
add a comment |
$begingroup$
This is a very simple, probably silly, question, but I'm having a diffuclty in unerstanding the terminology used in definition of a subspace.
From Wikipedia:
If $V$ is a vector space over a field $K$ and if $W$ is a subset of $V$, then
$W$ is a subspace of $V$ if under the operations of $V$, $W$ is a vector space
over $K$.
To me, the first sentence seems to be using two different (though related) terms interchangeably: "space" and a "subset".
In other words, why is it correct to say that "$W$ is subset of $V$", when $V$ is not merely a (super)set but a space? Am I being too pedantic over the phrasing?
linear-algebra vector-spaces
asked Dec 14 '18 at 13:29
HeyJudeHeyJude
1687
$endgroup$
1
$begingroup$
The words space and subspace have different meanings in different subjects. E.g. there are topological spaces and topological subspaces. In the context of vector spaces it is better to say vector subspace. Especially when talking about topological vector spaces. Actually $V$ is not a vector space, although we often do say that. A vector space over a field $Bbb F=(F,+_{Bbb F}, times_{Bbb F})$ is $Bbb V=(V,Bbb F, +,times)$ where $+$ is addition in $V$ and $times $ is multiplication of members of $V$ by members of $F$.
$endgroup$
– DanielWainfleet
Dec 14 '18 at 16:10
add a comment |
$begingroup$
This is a very simple, probably silly, question, but I'm having a diffuclty in unerstanding the terminology used in definition of a subspace.
From Wikipedia:
If $V$ is a vector space over a field $K$ and if $W$ is a subset of $V$, then
$W$ is a subspace of $V$ if under the operations of $V$, $W$ is a vector space
over $K$.
To me, the first sentence seems to be using two different (though related) terms interchangeably: "space" and a "subset".
In other words, why is it correct to say that "$W$ is subset of $V$", when $V$ is not merely a (super)set but a space? Am I being too pedantic over the phrasing?
linear-algebra vector-spaces
asked Dec 14 '18 at 13:29
HeyJudeHeyJude
1687
$endgroup$
This is a very simple, probably silly, question, but I'm having a diffuclty in unerstanding the terminology used in definition of a subspace.
From Wikipedia:
If $V$ is a vector space over a field $K$ and if $W$ is a subset of $V$, then
$W$ is a subspace of $V$ if under the operations of $V$, $W$ is a vector space
over $K$.
To me, the first sentence seems to be using two different (though related) terms interchangeably: "space" and a "subset".
In other words, why is it correct to say that "$W$ is subset of $V$", when $V$ is not merely a (super)set but a space? Am I being too pedantic over the phrasing?
linear-algebra vector-spaces
linear-algebra vector-spaces
asked Dec 14 '18 at 13:29
HeyJudeHeyJude
1687
asked Dec 14 '18 at 13:29
HeyJudeHeyJude
1687
asked Dec 14 '18 at 13:29
HeyJudeHeyJude
1687
asked Dec 14 '18 at 13:29
HeyJudeHeyJude
1687
asked Dec 14 '18 at 13:29
HeyJudeHeyJude
1687
1687
1
$begingroup$
The words space and subspace have different meanings in different subjects. E.g. there are topological spaces and topological subspaces. In the context of vector spaces it is better to say vector subspace. Especially when talking about topological vector spaces. Actually $V$ is not a vector space, although we often do say that. A vector space over a field $Bbb F=(F,+_{Bbb F}, times_{Bbb F})$ is $Bbb V=(V,Bbb F, +,times)$ where $+$ is addition in $V$ and $times $ is multiplication of members of $V$ by members of $F$.
$endgroup$
– DanielWainfleet
Dec 14 '18 at 16:10
add a comment |
1
$begingroup$
The words space and subspace have different meanings in different subjects. E.g. there are topological spaces and topological subspaces. In the context of vector spaces it is better to say vector subspace. Especially when talking about topological vector spaces. Actually $V$ is not a vector space, although we often do say that. A vector space over a field $Bbb F=(F,+_{Bbb F}, times_{Bbb F})$ is $Bbb V=(V,Bbb F, +,times)$ where $+$ is addition in $V$ and $times $ is multiplication of members of $V$ by members of $F$.
$endgroup$
– DanielWainfleet
Dec 14 '18 at 16:10
1
1
$begingroup$
The words space and subspace have different meanings in different subjects. E.g. there are topological spaces and topological subspaces. In the context of vector spaces it is better to say vector subspace. Especially when talking about topological vector spaces. Actually $V$ is not a vector space, although we often do say that. A vector space over a field $Bbb F=(F,+_{Bbb F}, times_{Bbb F})$ is $Bbb V=(V,Bbb F, +,times)$ where $+$ is addition in $V$ and $times $ is multiplication of members of $V$ by members of $F$.
$endgroup$
– DanielWainfleet
Dec 14 '18 at 16:10
$begingroup$
The words space and subspace have different meanings in different subjects. E.g. there are topological spaces and topological subspaces. In the context of vector spaces it is better to say vector subspace. Especially when talking about topological vector spaces. Actually $V$ is not a vector space, although we often do say that. A vector space over a field $Bbb F=(F,+_{Bbb F}, times_{Bbb F})$ is $Bbb V=(V,Bbb F, +,times)$ where $+$ is addition in $V$ and $times $ is multiplication of members of $V$ by members of $F$.
$endgroup$
– DanielWainfleet
Dec 14 '18 at 16:10
add a comment |
1 Answer
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$begingroup$
A subset of $V$ is any set that has no element not belonging to $V$. No matter what operations or relations $V$ has.
Now, if $W$ is a subset of $V$ and if the operations $+$ and $cdot$ inherited from $V$ make $W$ a vector space, then $W$ is a subspace of $V$.
For example, if $V=Bbb R^2$ and $W={(0,2),(1,4)}$ then $W$ is a subset of $V$, but it is not a subspace because $(0,2)+(1,4)notin W$.
answered Dec 14 '18 at 13:34
ajotatxeajotatxe
54.1k24190
$endgroup$
$begingroup$
So $V$ is a space and a (super)set at the same time, right?
$endgroup$
– HeyJude
Dec 14 '18 at 13:57
add a comment |
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$begingroup$
A subset of $V$ is any set that has no element not belonging to $V$. No matter what operations or relations $V$ has.
Now, if $W$ is a subset of $V$ and if the operations $+$ and $cdot$ inherited from $V$ make $W$ a vector space, then $W$ is a subspace of $V$.
For example, if $V=Bbb R^2$ and $W={(0,2),(1,4)}$ then $W$ is a subset of $V$, but it is not a subspace because $(0,2)+(1,4)notin W$.
answered Dec 14 '18 at 13:34
ajotatxeajotatxe
54.1k24190
$endgroup$
$begingroup$
So $V$ is a space and a (super)set at the same time, right?
$endgroup$
– HeyJude
Dec 14 '18 at 13:57
add a comment |
$begingroup$
A subset of $V$ is any set that has no element not belonging to $V$. No matter what operations or relations $V$ has.
Now, if $W$ is a subset of $V$ and if the operations $+$ and $cdot$ inherited from $V$ make $W$ a vector space, then $W$ is a subspace of $V$.
For example, if $V=Bbb R^2$ and $W={(0,2),(1,4)}$ then $W$ is a subset of $V$, but it is not a subspace because $(0,2)+(1,4)notin W$.
answered Dec 14 '18 at 13:34
ajotatxeajotatxe
54.1k24190
$endgroup$
$begingroup$
So $V$ is a space and a (super)set at the same time, right?
$endgroup$
– HeyJude
Dec 14 '18 at 13:57
add a comment |
$begingroup$
A subset of $V$ is any set that has no element not belonging to $V$. No matter what operations or relations $V$ has.
Now, if $W$ is a subset of $V$ and if the operations $+$ and $cdot$ inherited from $V$ make $W$ a vector space, then $W$ is a subspace of $V$.
For example, if $V=Bbb R^2$ and $W={(0,2),(1,4)}$ then $W$ is a subset of $V$, but it is not a subspace because $(0,2)+(1,4)notin W$.
answered Dec 14 '18 at 13:34
ajotatxeajotatxe
54.1k24190
$endgroup$
A subset of $V$ is any set that has no element not belonging to $V$. No matter what operations or relations $V$ has.
Now, if $W$ is a subset of $V$ and if the operations $+$ and $cdot$ inherited from $V$ make $W$ a vector space, then $W$ is a subspace of $V$.
For example, if $V=Bbb R^2$ and $W={(0,2),(1,4)}$ then $W$ is a subset of $V$, but it is not a subspace because $(0,2)+(1,4)notin W$.
answered Dec 14 '18 at 13:34
ajotatxeajotatxe
54.1k24190
answered Dec 14 '18 at 13:34
ajotatxeajotatxe
54.1k24190
answered Dec 14 '18 at 13:34
ajotatxeajotatxe
54.1k24190
answered Dec 14 '18 at 13:34
ajotatxeajotatxe
54.1k24190
54.1k24190
$begingroup$
So $V$ is a space and a (super)set at the same time, right?
$endgroup$
– HeyJude
Dec 14 '18 at 13:57
add a comment |
$begingroup$
So $V$ is a space and a (super)set at the same time, right?
$endgroup$
– HeyJude
Dec 14 '18 at 13:57
$begingroup$
So $V$ is a space and a (super)set at the same time, right?
$endgroup$
– HeyJude
Dec 14 '18 at 13:57
$begingroup$
So $V$ is a space and a (super)set at the same time, right?
$endgroup$
– HeyJude
Dec 14 '18 at 13:57
add a comment |
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$begingroup$
The words space and subspace have different meanings in different subjects. E.g. there are topological spaces and topological subspaces. In the context of vector spaces it is better to say vector subspace. Especially when talking about topological vector spaces. Actually $V$ is not a vector space, although we often do say that. A vector space over a field $Bbb F=(F,+_{Bbb F}, times_{Bbb F})$ is $Bbb V=(V,Bbb F, +,times)$ where $+$ is addition in $V$ and $times $ is multiplication of members of $V$ by members of $F$.
$endgroup$
– DanielWainfleet
Dec 14 '18 at 16:10