Theorems

Author: GeoffreySangston

theorems/T000877.md

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+---
+uid: T000877
+if:
+  and:
+  - P000238: true
+  - P000125: true
+then:
+  P000058: false
+---
+
+A nonzero real vector space contains a $1$-dimensional subspace, which is isomorphic to $\mathbb{R}$.

theorems/T000878.md

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+---
+uid: T000878
+if:
+  P000238: true
+then:
+  P000087: true
+---
+
+By definition.

theorems/T000879.md

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+---
+uid: T000879
+if:
+  P000238: true
+then:
+  P000199: true
+---
+
+$X \times [0, 1] \to X$, $(x, t) \mapsto (1-t)x$ is a null-homotopy of the identity map.

theorems/T000880.md

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+---
+uid: T000880
+if:
+  P000238: true
+then:
+  P000223: true
+refs:
+  - wikipedia: Balanced_set
+    name: Balanced set on Wikipedia
+  - zb: "0867.46001"
+    name: Functional analysis. 2nd ed. (W. Rudin)
+---
+
+Every neighborhood of the origin in a topological vector space $X$
+has a a balanced open neighborhood of the origin. See {{wikipedia:Balanced_set}}
+or Theorem 1.14(a) of {{zb:0867.4600}}.

theorems/T000881.md

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+---
+uid: T000881
+if:
+  and:
+  - P000238: true
+  - P000023: true
+  - P000003: true
+then:
+  P000124: true
+refs:
+  - wikipedia: F._Riesz%27s_theorem
+    name: F. Riesz's theorem on Wikipedia
+  - zb: "0867.46001"
+    name: Functional analysis. 2nd ed. (W. Rudin)
+---
+
+By F. Riesz's theorem, $X$ is homeomorphic to some Euclidean space $\mathbb{R}^n$.
+See {{wikipedia:F._Riesz%27s_theorem}} or Theorem 1.21 and Theorem 1.22 of
+{{zb:0867.46001}}. Euclidean spaces are topological manifolds by definition.