Riesz's lemma

Riesz's lemma states that for any proper closed subspace of a normed space and any number below one, a unit vector exists whose distance from the subspace exceeds that number.

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Quotient norm: The norm on a quotient space assigns each coset the infimum of the norms of its representatives. The distance from a vector to a subspace is its norm in the corresponding quotient space.
Riesz's theorem: Riesz's theorem characterizes locally compact Hausdorff topological vector spaces as finite-dimensional. Riesz's lemma supplies the separated vectors used to rule out local compactness in infinite dimensions.
Normed space: A normed space is a vector space equipped with a norm that measures vector length. The lemma needs a norm to define unit vectors and distances.
Bessel's inequality: Bessel's inequality bounds the sum of squared Fourier coefficients of a vector by its squared norm. Unlike Riesz's lemma, it concerns orthogonal projections and coefficient bounds in inner-product spaces.
Distance from a point to a subspace: The distance from a point to a subset is the infimum of its distances to points in that subset. The lemma constructs a unit vector whose distance from the closed subspace is large.
Compact operator: A compact operator maps bounded sets to sets with compact closure. Riesz's lemma helps show the identity operator on an infinite-dimensional normed space is not compact.
Linear subspace: A linear subspace is a subset of a vector space closed under addition and scalar multiplication. The lemma compares points with a proper linear subspace.
Orthogonal projection: An orthogonal projection maps a vector to its nearest point in a closed subspace of a Hilbert space. Riesz's lemma applies without an inner product or a nearest point in the subspace.
Infimum: The infimum of a set of real numbers is its greatest lower bound. The proof chooses a subspace element nearly attaining the infimum that defines distance.
Finite-dimensional normed space: A finite-dimensional normed space is a vector space of finite dimension equipped with a norm. In finite dimensions, bounded sequences have convergent subsequences, unlike the separated sequences the lemma permits.