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The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep ...
However, every finite dimensional normed space is a reflexive Banach space, so Riesz’s lemma does holds for = when the normed space is finite-dimensional, as will now be shown. When the dimension of X {\displaystyle X} is finite then the closed unit ball B ⊆ X {\displaystyle B\subseteq X} is compact.
In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman. [1] Let (X, ||·||) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds:
In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° (radians), or one of the vectors is zero. [4] Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension.
It was simply referred to as property (H) in a list of properties for normed spaces that starts with (A) and ends with (H). This list was given by K. Fan and I. Glicksberg (Observe that the definition of (H) given by Fan and Glicksberg includes additionally the rotundity of the norm, so it does not coincide with the Radon-Riesz property itself).
Conversely, the spaces H ℓ are precisely the eigenspaces of Δ S n−1. In particular, an application of the spectral theorem to the Riesz potential gives another proof that the spaces H ℓ are pairwise orthogonal and complete in L 2 (S n−1).
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace.Formally, starting with a linearly independent set of vectors {v 1, ... , v k} in an inner product space (most commonly the Euclidean space R n), orthogonalization results in a set of orthogonal vectors {u 1, ... , u k} that generate the same subspace as the vectors v 1 ...
A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". [1]