# new vector spaces from old ones

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The space of linear maps from one vector space to another, also denoted by $\operatorname{Hom}_{k}(V,W)$, or simply $\operatorname{Hom}(V,W)$, where $V$ and $W$ are vector spaces over the field $k$

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The space of endomorphisms  of a vector space. Using the notation above, this is the space $\operatorname{Hom}_{k}(V,V)=\operatorname{End}(V)$

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dual vector space (http://planetmath.org/DualSpace), and bi-dual vector space. Using the notation above, this is the space $\operatorname{Hom}(V,k)$, or simply $V^{*}$.

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A field $k$ is a vector space over itself. Consider a set $B$ and the set $V$ of all functions from $B$ to $k$. Then $V$ has a natural vector space structure. If $B$ is finite, then $V$ can be viewed as a vector space having $B$ as a basis.

## Vector spaces involving a linear map

Suppose $L\colon\thinspace V\to W$ is a linear map.

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The kernel of $L$ is a subspace of $V$.

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The image of $L$ is a subspace of $W$.

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## Topological vector spaces

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If $W$ is a subspace of $V$ then its closure  $\overline{W}$ is also a subspace of $V$.

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If $V$ is a metric vector space then its completion $\widetilde{V}$ is also a (metric) vector space.

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## Spaces of structures and subspaces of the tensor algebra of a vector space

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The space of Hermitian inner products.

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the space of symplectic structures.

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vector bundles

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space of connections

Title new vector spaces from old ones NewVectorSpacesFromOldOnes 2013-03-22 15:31:08 2013-03-22 15:31:08 matte (1858) matte (1858) 16 matte (1858) Topic msc 16-00 msc 13-00 msc 20-00 msc 15-00