A linear topological space over real number field is locally compact ??? The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Metric linear space and locally convex topological vector spaceIf both $H$ and $G/H$ are locally compact then $G$ is locally compact (topological Group)Must a complete space be locally compact?Locally Compact Topological Spaceis the vector space $mathbbR^mathbbN$ locally compact?Is every compact space locally compact?Non-Hausdorff locally compact topological fieldtopological vector space over locally compact fieldClarification on locally compact spaceProving every compact space is locally compact.
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A linear topological space over real number field is locally compact ???
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Metric linear space and locally convex topological vector spaceIf both $H$ and $G/H$ are locally compact then $G$ is locally compact (topological Group)Must a complete space be locally compact?Locally Compact Topological Spaceis the vector space $mathbbR^mathbbN$ locally compact?Is every compact space locally compact?Non-Hausdorff locally compact topological fieldtopological vector space over locally compact fieldClarification on locally compact spaceProving every compact space is locally compact.
$begingroup$
Let $V$ be a topological $mathbbR$-vector space with $rm dim(V) < infty$
Then, $V$ is locally compact $??$
general-topology vector-spaces compactness topological-vector-spaces locally-compact-groups
asked Mar 25 at 7:01
神宮寺春姫神宮寺春姫
494
$endgroup$
add a comment |
$begingroup$
Let $V$ be a topological $mathbbR$-vector space with $rm dim(V) < infty$
Then, $V$ is locally compact $??$
general-topology vector-spaces compactness topological-vector-spaces locally-compact-groups
asked Mar 25 at 7:01
神宮寺春姫神宮寺春姫
494
$endgroup$
add a comment |
$begingroup$
Let $V$ be a topological $mathbbR$-vector space with $rm dim(V) < infty$
Then, $V$ is locally compact $??$
general-topology vector-spaces compactness topological-vector-spaces locally-compact-groups
asked Mar 25 at 7:01
神宮寺春姫神宮寺春姫
494
$endgroup$
Let $V$ be a topological $mathbbR$-vector space with $rm dim(V) < infty$
Then, $V$ is locally compact $??$
general-topology vector-spaces compactness topological-vector-spaces locally-compact-groups
general-topology vector-spaces compactness topological-vector-spaces locally-compact-groups
asked Mar 25 at 7:01
神宮寺春姫神宮寺春姫
494
asked Mar 25 at 7:01
神宮寺春姫神宮寺春姫
494
asked Mar 25 at 7:01
神宮寺春姫神宮寺春姫
494
asked Mar 25 at 7:01
神宮寺春姫神宮寺春姫
494
asked Mar 25 at 7:01
神宮寺春姫神宮寺春姫
494
494
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
A standard theorem in topological vector spaces (over the reals) says that if $n = dim(V) < infty$ then $V simeq mathbbR^n$, so in particular it is locally compact. This is non-trivial, and the converse also holds: if the dimension is not finite, then $V$ is not locally compact.
E.g. See theorems 1.20 and 1.21 in Rudin's Functional Analysis.
Or pages 244-245 of Dunford and Schwartz Linear Operators Part I, General Theory. To name the first two books on functional analysis on my shelf I tried. It's in most standard texts.
answered Mar 25 at 7:05
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Are there references which contain the theorem and its proof ??
$endgroup$
– 神宮寺春姫
Mar 25 at 7:12
$begingroup$
@神宮寺春姫 any good book on functional analysis? Use Google?
$endgroup$
– Henno Brandsma
Mar 25 at 7:13
$begingroup$
ok. I try to find it by google.
$endgroup$
– 神宮寺春姫
Mar 25 at 7:18
add a comment |
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1 Answer
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1 Answer
1
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active
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$begingroup$
A standard theorem in topological vector spaces (over the reals) says that if $n = dim(V) < infty$ then $V simeq mathbbR^n$, so in particular it is locally compact. This is non-trivial, and the converse also holds: if the dimension is not finite, then $V$ is not locally compact.
E.g. See theorems 1.20 and 1.21 in Rudin's Functional Analysis.
Or pages 244-245 of Dunford and Schwartz Linear Operators Part I, General Theory. To name the first two books on functional analysis on my shelf I tried. It's in most standard texts.
answered Mar 25 at 7:05
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Are there references which contain the theorem and its proof ??
$endgroup$
– 神宮寺春姫
Mar 25 at 7:12
$begingroup$
@神宮寺春姫 any good book on functional analysis? Use Google?
$endgroup$
– Henno Brandsma
Mar 25 at 7:13
$begingroup$
ok. I try to find it by google.
$endgroup$
– 神宮寺春姫
Mar 25 at 7:18
add a comment |
$begingroup$
A standard theorem in topological vector spaces (over the reals) says that if $n = dim(V) < infty$ then $V simeq mathbbR^n$, so in particular it is locally compact. This is non-trivial, and the converse also holds: if the dimension is not finite, then $V$ is not locally compact.
E.g. See theorems 1.20 and 1.21 in Rudin's Functional Analysis.
Or pages 244-245 of Dunford and Schwartz Linear Operators Part I, General Theory. To name the first two books on functional analysis on my shelf I tried. It's in most standard texts.
answered Mar 25 at 7:05
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Are there references which contain the theorem and its proof ??
$endgroup$
– 神宮寺春姫
Mar 25 at 7:12
$begingroup$
@神宮寺春姫 any good book on functional analysis? Use Google?
$endgroup$
– Henno Brandsma
Mar 25 at 7:13
$begingroup$
ok. I try to find it by google.
$endgroup$
– 神宮寺春姫
Mar 25 at 7:18
add a comment |
$begingroup$
A standard theorem in topological vector spaces (over the reals) says that if $n = dim(V) < infty$ then $V simeq mathbbR^n$, so in particular it is locally compact. This is non-trivial, and the converse also holds: if the dimension is not finite, then $V$ is not locally compact.
E.g. See theorems 1.20 and 1.21 in Rudin's Functional Analysis.
Or pages 244-245 of Dunford and Schwartz Linear Operators Part I, General Theory. To name the first two books on functional analysis on my shelf I tried. It's in most standard texts.
answered Mar 25 at 7:05
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
A standard theorem in topological vector spaces (over the reals) says that if $n = dim(V) < infty$ then $V simeq mathbbR^n$, so in particular it is locally compact. This is non-trivial, and the converse also holds: if the dimension is not finite, then $V$ is not locally compact.
E.g. See theorems 1.20 and 1.21 in Rudin's Functional Analysis.
Or pages 244-245 of Dunford and Schwartz Linear Operators Part I, General Theory. To name the first two books on functional analysis on my shelf I tried. It's in most standard texts.
answered Mar 25 at 7:05
Henno BrandsmaHenno Brandsma
116k349127
edited Mar 25 at 9:03
answered Mar 25 at 7:05
Henno BrandsmaHenno Brandsma
116k349127
answered Mar 25 at 7:05
Henno BrandsmaHenno Brandsma
116k349127
answered Mar 25 at 7:05
Henno BrandsmaHenno Brandsma
116k349127
116k349127
$begingroup$
Are there references which contain the theorem and its proof ??
$endgroup$
– 神宮寺春姫
Mar 25 at 7:12
$begingroup$
@神宮寺春姫 any good book on functional analysis? Use Google?
$endgroup$
– Henno Brandsma
Mar 25 at 7:13
$begingroup$
ok. I try to find it by google.
$endgroup$
– 神宮寺春姫
Mar 25 at 7:18
add a comment |
$begingroup$
Are there references which contain the theorem and its proof ??
$endgroup$
– 神宮寺春姫
Mar 25 at 7:12
$begingroup$
@神宮寺春姫 any good book on functional analysis? Use Google?
$endgroup$
– Henno Brandsma
Mar 25 at 7:13
$begingroup$
ok. I try to find it by google.
$endgroup$
– 神宮寺春姫
Mar 25 at 7:18
$begingroup$
Are there references which contain the theorem and its proof ??
$endgroup$
– 神宮寺春姫
Mar 25 at 7:12
$begingroup$
Are there references which contain the theorem and its proof ??
$endgroup$
– 神宮寺春姫
Mar 25 at 7:12
$begingroup$
@神宮寺春姫 any good book on functional analysis? Use Google?
$endgroup$
– Henno Brandsma
Mar 25 at 7:13
$begingroup$
@神宮寺春姫 any good book on functional analysis? Use Google?
$endgroup$
– Henno Brandsma
Mar 25 at 7:13
$begingroup$
ok. I try to find it by google.
$endgroup$
– 神宮寺春姫
Mar 25 at 7:18
$begingroup$
ok. I try to find it by google.
$endgroup$
– 神宮寺春姫
Mar 25 at 7:18
add a comment |
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