What's the definition of proper subspace of a vector space used in Rudin's Functional analysis
$begingroup$
I'm reading through the Rudin's functional analysis, and theorem 3.5 use the term "Proper Subspace", there's a theorem in chapter 2 that uses the same terminology.
I'm reading through chapter 1 again, and through the glossary as well but I cannot find the definition used.
I guess there must be a standard definition then,
What is such definition?
functional-analysis vector-spaces definition topological-vector-spaces
asked Nov 30 '18 at 10:25
user8469759user8469759
1,3851617
$endgroup$
add a comment |
$begingroup$
I'm reading through the Rudin's functional analysis, and theorem 3.5 use the term "Proper Subspace", there's a theorem in chapter 2 that uses the same terminology.
I'm reading through chapter 1 again, and through the glossary as well but I cannot find the definition used.
I guess there must be a standard definition then,
What is such definition?
functional-analysis vector-spaces definition topological-vector-spaces
asked Nov 30 '18 at 10:25
user8469759user8469759
1,3851617
$endgroup$
2
$begingroup$
It a subspace different from the vector space.
$endgroup$
– Bernard
Nov 30 '18 at 10:29
$begingroup$
So literally it's a proper subset that is also a vector space, correct?
$endgroup$
– user8469759
Nov 30 '18 at 10:30
$begingroup$
Yes, absolutely.
$endgroup$
– Bernard
Nov 30 '18 at 10:33
add a comment |
$begingroup$
I'm reading through the Rudin's functional analysis, and theorem 3.5 use the term "Proper Subspace", there's a theorem in chapter 2 that uses the same terminology.
I'm reading through chapter 1 again, and through the glossary as well but I cannot find the definition used.
I guess there must be a standard definition then,
What is such definition?
functional-analysis vector-spaces definition topological-vector-spaces
asked Nov 30 '18 at 10:25
user8469759user8469759
1,3851617
$endgroup$
I'm reading through the Rudin's functional analysis, and theorem 3.5 use the term "Proper Subspace", there's a theorem in chapter 2 that uses the same terminology.
I'm reading through chapter 1 again, and through the glossary as well but I cannot find the definition used.
I guess there must be a standard definition then,
What is such definition?
functional-analysis vector-spaces definition topological-vector-spaces
functional-analysis vector-spaces definition topological-vector-spaces
asked Nov 30 '18 at 10:25
user8469759user8469759
1,3851617
asked Nov 30 '18 at 10:25
user8469759user8469759
1,3851617
asked Nov 30 '18 at 10:25
user8469759user8469759
1,3851617
asked Nov 30 '18 at 10:25
user8469759user8469759
1,3851617
asked Nov 30 '18 at 10:25
user8469759user8469759
1,3851617
1,3851617
2
$begingroup$
It a subspace different from the vector space.
$endgroup$
– Bernard
Nov 30 '18 at 10:29
$begingroup$
So literally it's a proper subset that is also a vector space, correct?
$endgroup$
– user8469759
Nov 30 '18 at 10:30
$begingroup$
Yes, absolutely.
$endgroup$
– Bernard
Nov 30 '18 at 10:33
add a comment |
2
$begingroup$
It a subspace different from the vector space.
$endgroup$
– Bernard
Nov 30 '18 at 10:29
$begingroup$
So literally it's a proper subset that is also a vector space, correct?
$endgroup$
– user8469759
Nov 30 '18 at 10:30
$begingroup$
Yes, absolutely.
$endgroup$
– Bernard
Nov 30 '18 at 10:33
2
2
$begingroup$
It a subspace different from the vector space.
$endgroup$
– Bernard
Nov 30 '18 at 10:29
$begingroup$
It a subspace different from the vector space.
$endgroup$
– Bernard
Nov 30 '18 at 10:29
$begingroup$
So literally it's a proper subset that is also a vector space, correct?
$endgroup$
– user8469759
Nov 30 '18 at 10:30
$begingroup$
So literally it's a proper subset that is also a vector space, correct?
$endgroup$
– user8469759
Nov 30 '18 at 10:30
$begingroup$
Yes, absolutely.
$endgroup$
– Bernard
Nov 30 '18 at 10:33
$begingroup$
Yes, absolutely.
$endgroup$
– Bernard
Nov 30 '18 at 10:33
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$V$ is a proper subspace of $X$ if
$V$ is a subspace of $X$ and $Vsubsetneq X$.
I would guess that there is no definition of proper subspace in the book,
since a proper subspace is a subspace that is also a proper subset.
answered Nov 30 '18 at 10:29
supinfsupinf
6,1241028
$endgroup$
$begingroup$
Does this definition imply that the only proper subspace of $mathbb{R}$ is $left{ 0 right}$ and that proper subspaces of topological vector spaces have empty interiors?
$endgroup$
– user8469759
Nov 30 '18 at 10:35
$begingroup$
@user8469759 yes
$endgroup$
– supinf
Nov 30 '18 at 10:35
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$V$ is a proper subspace of $X$ if
$V$ is a subspace of $X$ and $Vsubsetneq X$.
I would guess that there is no definition of proper subspace in the book,
since a proper subspace is a subspace that is also a proper subset.
answered Nov 30 '18 at 10:29
supinfsupinf
6,1241028
$endgroup$
$begingroup$
Does this definition imply that the only proper subspace of $mathbb{R}$ is $left{ 0 right}$ and that proper subspaces of topological vector spaces have empty interiors?
$endgroup$
– user8469759
Nov 30 '18 at 10:35
$begingroup$
@user8469759 yes
$endgroup$
– supinf
Nov 30 '18 at 10:35
add a comment |
$begingroup$
$V$ is a proper subspace of $X$ if
$V$ is a subspace of $X$ and $Vsubsetneq X$.
I would guess that there is no definition of proper subspace in the book,
since a proper subspace is a subspace that is also a proper subset.
answered Nov 30 '18 at 10:29
supinfsupinf
6,1241028
$endgroup$
$begingroup$
Does this definition imply that the only proper subspace of $mathbb{R}$ is $left{ 0 right}$ and that proper subspaces of topological vector spaces have empty interiors?
$endgroup$
– user8469759
Nov 30 '18 at 10:35
$begingroup$
@user8469759 yes
$endgroup$
– supinf
Nov 30 '18 at 10:35
add a comment |
$begingroup$
$V$ is a proper subspace of $X$ if
$V$ is a subspace of $X$ and $Vsubsetneq X$.
I would guess that there is no definition of proper subspace in the book,
since a proper subspace is a subspace that is also a proper subset.
answered Nov 30 '18 at 10:29
supinfsupinf
6,1241028
$endgroup$
$V$ is a proper subspace of $X$ if
$V$ is a subspace of $X$ and $Vsubsetneq X$.
I would guess that there is no definition of proper subspace in the book,
since a proper subspace is a subspace that is also a proper subset.
answered Nov 30 '18 at 10:29
supinfsupinf
6,1241028
edited Nov 30 '18 at 10:34
answered Nov 30 '18 at 10:29
supinfsupinf
6,1241028
answered Nov 30 '18 at 10:29
supinfsupinf
6,1241028
answered Nov 30 '18 at 10:29
supinfsupinf
6,1241028
6,1241028
$begingroup$
Does this definition imply that the only proper subspace of $mathbb{R}$ is $left{ 0 right}$ and that proper subspaces of topological vector spaces have empty interiors?
$endgroup$
– user8469759
Nov 30 '18 at 10:35
$begingroup$
@user8469759 yes
$endgroup$
– supinf
Nov 30 '18 at 10:35
add a comment |
$begingroup$
Does this definition imply that the only proper subspace of $mathbb{R}$ is $left{ 0 right}$ and that proper subspaces of topological vector spaces have empty interiors?
$endgroup$
– user8469759
Nov 30 '18 at 10:35
$begingroup$
@user8469759 yes
$endgroup$
– supinf
Nov 30 '18 at 10:35
$begingroup$
Does this definition imply that the only proper subspace of $mathbb{R}$ is $left{ 0 right}$ and that proper subspaces of topological vector spaces have empty interiors?
$endgroup$
– user8469759
Nov 30 '18 at 10:35
$begingroup$
Does this definition imply that the only proper subspace of $mathbb{R}$ is $left{ 0 right}$ and that proper subspaces of topological vector spaces have empty interiors?
$endgroup$
– user8469759
Nov 30 '18 at 10:35
$begingroup$
@user8469759 yes
$endgroup$
– supinf
Nov 30 '18 at 10:35
$begingroup$
@user8469759 yes
$endgroup$
– supinf
Nov 30 '18 at 10:35
add a comment |
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2
$begingroup$
It a subspace different from the vector space.
$endgroup$
– Bernard
Nov 30 '18 at 10:29
$begingroup$
So literally it's a proper subset that is also a vector space, correct?
$endgroup$
– user8469759
Nov 30 '18 at 10:30
$begingroup$
Yes, absolutely.
$endgroup$
– Bernard
Nov 30 '18 at 10:33