What's the definition of proper subspace of a vector space used in Rudin's Functional analysis
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 at 10:25
user8469759
1,3311616
add a comment |
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 at 10:25
user8469759
1,3311616
2
It a subspace different from the vector space.
– Bernard
Nov 30 at 10:29
So literally it's a proper subset that is also a vector space, correct?
– user8469759
Nov 30 at 10:30
Yes, absolutely.
– Bernard
Nov 30 at 10:33
add a comment |
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 at 10:25
user8469759
1,3311616
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 at 10:25
user8469759
1,3311616
asked Nov 30 at 10:25
user8469759
1,3311616
asked Nov 30 at 10:25
user8469759
1,3311616
asked Nov 30 at 10:25
user8469759
1,3311616
asked Nov 30 at 10:25
user8469759
1,3311616
1,3311616
2
It a subspace different from the vector space.
– Bernard
Nov 30 at 10:29
So literally it's a proper subset that is also a vector space, correct?
– user8469759
Nov 30 at 10:30
Yes, absolutely.
– Bernard
Nov 30 at 10:33
add a comment |
2
It a subspace different from the vector space.
– Bernard
Nov 30 at 10:29
So literally it's a proper subset that is also a vector space, correct?
– user8469759
Nov 30 at 10:30
Yes, absolutely.
– Bernard
Nov 30 at 10:33
2
2
It a subspace different from the vector space.
– Bernard
Nov 30 at 10:29
It a subspace different from the vector space.
– Bernard
Nov 30 at 10:29
So literally it's a proper subset that is also a vector space, correct?
– user8469759
Nov 30 at 10:30
So literally it's a proper subset that is also a vector space, correct?
– user8469759
Nov 30 at 10:30
Yes, absolutely.
– Bernard
Nov 30 at 10:33
Yes, absolutely.
– Bernard
Nov 30 at 10:33
add a comment |
1 Answer
1
active
oldest
votes
$V$ 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 at 10:29
supinf
5,9491027
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?
– user8469759
Nov 30 at 10:35
@user8469759 yes
– supinf
Nov 30 at 10:35
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$V$ 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 at 10:29
supinf
5,9491027
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?
– user8469759
Nov 30 at 10:35
@user8469759 yes
– supinf
Nov 30 at 10:35
add a comment |
$V$ 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 at 10:29
supinf
5,9491027
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?
– user8469759
Nov 30 at 10:35
@user8469759 yes
– supinf
Nov 30 at 10:35
add a comment |
$V$ 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 at 10:29
supinf
5,9491027
$V$ 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 at 10:29
supinf
5,9491027
edited Nov 30 at 10:34
answered Nov 30 at 10:29
supinf
5,9491027
answered Nov 30 at 10:29
supinf
5,9491027
answered Nov 30 at 10:29
supinf
5,9491027
5,9491027
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?
– user8469759
Nov 30 at 10:35
@user8469759 yes
– supinf
Nov 30 at 10:35
add a comment |
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?
– user8469759
Nov 30 at 10:35
@user8469759 yes
– supinf
Nov 30 at 10:35
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?
– user8469759
Nov 30 at 10:35
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?
– user8469759
Nov 30 at 10:35
@user8469759 yes
– supinf
Nov 30 at 10:35
@user8469759 yes
– supinf
Nov 30 at 10:35
add a comment |
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2
It a subspace different from the vector space.
– Bernard
Nov 30 at 10:29
So literally it's a proper subset that is also a vector space, correct?
– user8469759
Nov 30 at 10:30
Yes, absolutely.
– Bernard
Nov 30 at 10:33