| Version 16 |
Version 15 |
| \PMlinkescapeword{continuous} |
\PMlinkescapeword{continuous} |
| \PMlinkescapeword{equivalent} |
\PMlinkescapeword{equivalent} |
| \PMlinkescapeword{independent} |
\PMlinkescapeword{independent} |
| \PMlinkescapeword{inverse} |
\PMlinkescapeword{inverse} |
| \PMlinkescapeword{scalar} |
\PMlinkescapeword{scalar} |
| \PMlinkescapeword{vector} |
\PMlinkescapeword{vector} |
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| A \emph{topological vector space} is a pair $(V,\mathcal{T})$, |
A \emph{topological vector space} is a pair $(V,\mathcal{T})$, |
| where $V$ is a vector space over a topological field $K$, |
where $V$ is a vector space over a topological field $K$, |
| and $\mathcal{T}$ is a topology on $V$ such that under $\mathcal{T}$ |
and $\mathcal{T}$ is a topology on $V$ such that under $\mathcal{T}$ |
| the scalar multiplication $(\lambda,v)\mapsto\lambda v$ |
the scalar multiplication $(\lambda,v)\mapsto\lambda v$ |
| is a continuous function $K\times V\to V$ |
is a continuous function $K\times V\to V$ |
| and the vector addition $(v,w)\mapsto v+w$ |
and the vector addition $(v,w)\mapsto v+w$ |
| is a continuous function $V\times V\to V$, |
is a continuous function $V\times V\to V$, |
| where $K\times V$ and $V\times V$ are given the respective product topologies. |
where $K\times V$ and $V\times V$ are given the respective product topologies. |
| We will also require that $\{0\}$ is closed |
We will also require that $\{0\}$ is closed |
| (which is equivalent to requiring the topology to be Hausdorff), |
(which is equivalent to requiring the topology to be Hausdorff), |
| though some authors do not make this requirement. |
though some authors do not make this requirement. |
| Many authors require that $K$ be either $\R$ or $\C$ |
Many authors require that $K$ be either $\R$ or $\C$ |
| (with their usual topologies). |
(with their usual topologies). |
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| A topological vector space is necessarily a topological group: |
A topological vector space is necessarily a topological group: |
| the definition ensures that the group operation (vector addition) is continuous, |
the definition ensures that the group operation (vector addition) is continuous, |
| and the inverse operation is the same as multiplication by $-1$, |
and the inverse operation is the same as multiplication by $-1$, |
| and so is also continuous. |
and so is also continuous. |
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| A finite dimensional vector space inherits a natural topology. For if $V$ is a finite dimensional vector space, then $V$ is isomorphic to $K^n$ for some $n$; then let $f\colon V\rightarrow K^n$ be such an isomorphism, and suppose $K^n$ has the product topology. Give $V$ the topology where a subset $A$ of $V$ is open in $V$ if and only if $f(A)$ is open in $K^n$. This topology is independent of the choice of isomorphism $f$. |
A finite dimensional vector space inherits a natural topology. For if $V$ is a finite dimensional vector space, then $V$ is isomorphic to $K^n$ for some $n$; then let $f\colon V\rightarrow K^n$ be such an isomorphism, and suppose $K^n$ has the product topology. Give $V$ the topology where a subset $A$ of $V$ is open in $V$ if and only if $f(A)$ is open in $K^n$. This topology is independent of the choice of isomorphism $f$. |
