Mixing
Cardinality of infinite dimensional vector space
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Assume that V is an infinite dimensional vector space. I know that if V is a vector space over a field F, then |V|=max{dimV,|F|}. So if we take V=$mathbb{R}$ and F=$mathbb{Q}$ then |V|>|F| and |V|=dimV (Cardinality of a basis of an infinite-dimensional vector space).
Is there any example for the case |V|>dimV and |V|=|F|?
linear-algebra cardinals
asked Jan 22 at 20:22
uio666uio666
62
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add a comment |
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Assume that V is an infinite dimensional vector space. I know that if V is a vector space over a field F, then |V|=max{dimV,|F|}. So if we take V=$mathbb{R}$ and F=$mathbb{Q}$ then |V|>|F| and |V|=dimV (Cardinality of a basis of an infinite-dimensional vector space).
Is there any example for the case |V|>dimV and |V|=|F|?
linear-algebra cardinals
asked Jan 22 at 20:22
uio666uio666
62
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$dim V=1{{}}$?
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– Lord Shark the Unknown
Jan 22 at 20:27
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@LordSharktheUnknown The title says infinite dimensional vector spaces. I don't know if it applies to the desired example. We will see. I would suggest $mathbb R^mathbb N$ over $mathbb R$.
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– Dog_69
Jan 22 at 22:16
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How that works?
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– uio666
Jan 23 at 6:12
add a comment |
$begingroup$
Assume that V is an infinite dimensional vector space. I know that if V is a vector space over a field F, then |V|=max{dimV,|F|}. So if we take V=$mathbb{R}$ and F=$mathbb{Q}$ then |V|>|F| and |V|=dimV (Cardinality of a basis of an infinite-dimensional vector space).
Is there any example for the case |V|>dimV and |V|=|F|?
linear-algebra cardinals
asked Jan 22 at 20:22
uio666uio666
62
$endgroup$
Assume that V is an infinite dimensional vector space. I know that if V is a vector space over a field F, then |V|=max{dimV,|F|}. So if we take V=$mathbb{R}$ and F=$mathbb{Q}$ then |V|>|F| and |V|=dimV (Cardinality of a basis of an infinite-dimensional vector space).
Is there any example for the case |V|>dimV and |V|=|F|?
linear-algebra cardinals
linear-algebra cardinals
asked Jan 22 at 20:22
uio666uio666
62
asked Jan 22 at 20:22
uio666uio666
62
edited Jan 23 at 6:11
uio666
asked Jan 22 at 20:22
uio666uio666
62
asked Jan 22 at 20:22
uio666uio666
62
asked Jan 22 at 20:22
uio666uio666
62
62
$begingroup$
$dim V=1{{}}$?
$endgroup$
– Lord Shark the Unknown
Jan 22 at 20:27
$begingroup$
@LordSharktheUnknown The title says infinite dimensional vector spaces. I don't know if it applies to the desired example. We will see. I would suggest $mathbb R^mathbb N$ over $mathbb R$.
$endgroup$
– Dog_69
Jan 22 at 22:16
$begingroup$
How that works?
$endgroup$
– uio666
Jan 23 at 6:12
add a comment |
$begingroup$
$dim V=1{{}}$?
$endgroup$
– Lord Shark the Unknown
Jan 22 at 20:27
$begingroup$
@LordSharktheUnknown The title says infinite dimensional vector spaces. I don't know if it applies to the desired example. We will see. I would suggest $mathbb R^mathbb N$ over $mathbb R$.
$endgroup$
– Dog_69
Jan 22 at 22:16
$begingroup$
How that works?
$endgroup$
– uio666
Jan 23 at 6:12
$begingroup$
$dim V=1{{}}$?
$endgroup$
– Lord Shark the Unknown
Jan 22 at 20:27
$begingroup$
$dim V=1{{}}$?
$endgroup$
– Lord Shark the Unknown
Jan 22 at 20:27
$begingroup$
@LordSharktheUnknown The title says infinite dimensional vector spaces. I don't know if it applies to the desired example. We will see. I would suggest $mathbb R^mathbb N$ over $mathbb R$.
$endgroup$
– Dog_69
Jan 22 at 22:16
$begingroup$
@LordSharktheUnknown The title says infinite dimensional vector spaces. I don't know if it applies to the desired example. We will see. I would suggest $mathbb R^mathbb N$ over $mathbb R$.
$endgroup$
– Dog_69
Jan 22 at 22:16
$begingroup$
How that works?
$endgroup$
– uio666
Jan 23 at 6:12
$begingroup$
How that works?
$endgroup$
– uio666
Jan 23 at 6:12
add a comment |
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$begingroup$
$dim V=1{{}}$?
$endgroup$
– Lord Shark the Unknown
Jan 22 at 20:27
$begingroup$
@LordSharktheUnknown The title says infinite dimensional vector spaces. I don't know if it applies to the desired example. We will see. I would suggest $mathbb R^mathbb N$ over $mathbb R$.
$endgroup$
– Dog_69
Jan 22 at 22:16
$begingroup$
How that works?
$endgroup$
– uio666
Jan 23 at 6:12