normsleft

Normsleft is a term used in functional analysis to describe a family of left-projected norms on a normed or Banach space. The concept arises from selecting a fixed closed subspace L of a space X and a complementary subspace M such that X = L ⊕ M. For each element x in X, one obtains a unique decomposition x = l + m with l ∈ L and m ∈ M. The left norm of x, denoted normsleft(x), is defined as the norm of the L-component, typically normsleft(x) = ||l||.

In general, normsleft is a semi-norm, because different vectors can share the same left component. It becomes a genuine norm when the decomposition is topologically exact, i.e., the projection onto L is injective on the space considered, or equivalently L ∩ M = {0} with a proper decomposition. Variants may introduce a parameterized family normsleft_t(x) = ||l|| + t||m|| for t in [0,1], which interpolate between focusing on the left component and incorporating part of the right component.

Properties and relationships: normsleft is subadditive and positively homogeneous, inheriting these basic norm-like properties from ||·|| on X. If the projection onto L is continuous, normsleft is continuous with respect to the original norm. The construction is closely related to projection-based decompositions and the study of how partial information (the left component) controls the size of vectors.

Applications: Normsleft is used in decomposition-based techniques in numerical linear algebra, signal processing, and approximation theory, where emphasis on a particular subspace component is desirable. It can aid in designing preconditioners, sparse representations, or in analysis where control of a subspace component is relevant.

See also: norm, Banach space, projection, semi-norm, decomposition of a vector space.