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Closed finite-codimensional subspaces are complemented
Statement
Every closed finite-codimensional linear subspace of a normed space is complemented in .
Facts & Assumptions
Given: A closed subspace with finite-dimensional .
The quotient seminorm is a norm exactly when the subspace is closed (The quotient seminorm is a norm exactly when the subspace is closed).
Coordinate functionals for a fixed basis of a finite-dimensional normed space are bounded (A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space).
The range of a bounded projection is complemented (A closed subspace is complemented exactly when it is the range of a bounded projection).
Proof
By [F1], is a normed finite-dimensional quotient. Choose a basis of , representatives , and bounded coordinate maps supplied by [F2].
The map , , is bounded and satisfies . Hence is bounded, , and .
By [F3], is complemented.
Depends on
The quotient seminorm (|x+M|{X/M}=\inf{m\in M}|x+m|=\operatorname{dist}(x,M))
The quotient seminorm is a norm exactly when the subspace is closed
A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space
A complemented closed subspace of a normed space
A closed subspace is complemented exactly when it is the range of a bounded projection