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A linear subspace that is closed in the topology induced by the ambient norm.
A linear subspace of a normed vector space is closed if it contains the limit of every convergent sequence of points of . Equivalently, its complement is open in the subspace topology inherited from .
A closed linear subspace of a Banach space is again a Banach space. In a Hilbert space, every closed subspace has an orthogonal complement and every vector decomposes uniquely as .
Let be a vector space over , and let .
The set is a linear subspace of if:
With the inherited operations, is itself a vector space, and many constructions in analysis arise as subspaces (kernels, images, function spaces, etc.). Subspaces are also the building blocks for quotient spaces and direct sums.
A normed vector space is a vector space equipped with a norm on .
A Banach space is a normed vector space such that every Cauchy sequence in converges (in the norm) to a point of .
A Hilbert space is an inner product space that is complete with respect to the induced norm .
For a subset of an inner-product space , its orthogonal complement is
It is always a closed linear subspace. If is closed in a Hilbert space, then every has a unique decomposition with and , so .