Example: Trivial Subspaces, Zero Space

Please note that every vector space $V$ over a given field $F$ has two trivial subspaces:

$V$ itself is a subspace of $V$ , and
The subset $U:=\{0_V\}$ consisting of the zero vector of $V$ is also a subspace of $V$ .

$U$ is sometimes also called the zero space. The vector $0_V$ has to exist, since $(V,\oplus)$ is a group, which is part of the definition of vector space. The index of the vector $0_V$ is used to differentiate it from the zero element of the addition $0\in F$ , and can be omitted, if it is clear from the context, which zero element is meant.

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Example: Trivial Subspaces, Zero Space