Normed spaces, completions and quotients

A norm is positive definite, absolutely homogeneous and subadditive. It gives a translation-invariant metric by taking the norm of a difference. A Banach space is a normed space in which every Cauchy sequence converges in that norm. A closed linear subspace of a Banach space is Banach, while a dense proper subspace is incomplete in the inherited norm. Absolute convergence of a vector series, meaning summability of its term norms, guarantees convergence in a Banach space by the Cauchy criterion; without completeness the limit may be missing.

Two equivalent norms bound one another by fixed positive constants, so they give the same Cauchy sequences, topology and completeness. All norms on a fixed finite-dimensional space are equivalent, but the constants can depend on dimension. Infinite-dimensional analogues need not hold. On finitely supported sequences, the supremum norm is at most the summation norm, but no reverse constant works uniformly: a vector with its first N entries equal to one has supremum norm one and summation norm N. A formal similarity between two norm formulas therefore does not establish equivalent geometry.

A completion is a Banach space containing an isometric copy of the original normed space as a dense subspace. One constructs it from Cauchy sequences, identifying two sequences when their difference tends to zero. Addition and scalar multiplication pass to equivalence classes, and the limit of the original term norms defines the new norm. The original vector embeds as its constant sequence. Every completion is uniquely isometrically isomorphic to any other through a map fixing this embedded copy, so the construction does not depend on arbitrary choices of sequence representatives.

Let c₀₀ be the finitely supported scalar sequences. Its completion in the supremum norm is c₀, the sequences tending to zero, because truncation converges uniformly exactly when the tail supremum tends to zero. In the square-summation norm its completion is ℓ² instead. These spaces contain different vectors and have different duality behavior. A bounded linear map from a normed space into a Banach space extends uniquely to its completion by sending a Cauchy representative to the limit of its images; the operator norm remains unchanged.

For a linear subspace M of X, a coset x+M records all vectors differing from x by an element of M. Its natural size is the infimum of the norms of its representatives. This expression is always a seminorm, but it is a genuine norm precisely when M is closed. Otherwise a vector in the closure of M but outside M produces a nonzero coset of size zero. The correct normed quotient in that situation removes the closure instead. Infima need not be attained by a representative in a general Banach space.

If X is Banach and M is closed, the quotient is Banach. A useful proof takes a rapidly Cauchy subsequence of cosets and chooses representatives of successive differences with summable norms. Completeness of X sums these corrections, and the quotient map carries their limit to the desired quotient limit. This proof uses approximate minimizing representatives rather than a nearest-point projection. Orthogonal projection supplies extra structure in Hilbert spaces, but it should not be imported into arbitrary Banach quotients, where a closest representative can fail to exist.