^^Spazio dotato di prodotto interno.

Spazio vettoriale dotato di prodotto interno, sul campo reale

Parallelogram law

The norm associated with any inner product space satisfies the parallelogram law:

‖x+y‖² + ‖x−y‖² = 2‖x‖² + 2‖y‖² 

In fact, as observed by John von Neumann, the parallelogram law characterizes those norms that arise from inner products.

wp/Polarization_identity

wp/Inner_product_space | wp/Definite_quadratic_form

Teo: A linear map V→W between two finite dimensional inner product spaces

• is an isomorphism of inner product spaces
• if and only if the image of a (some fixed) orthonomal basis of V is an orthonormal basis of W.

Teo: If V is a real vector space of dimension n, and B a basis of V, then
there is a unique inner product on V for which this basis is orthonormal.

cmt: With f_B:V→Rⁿ the map sending a vector v to its coordinates in B, this inner product is obtained from the by transport of structure via f_B, in other words ⟨v,w⟩_V = ⟨fB(v),fB(w)⟩_Rⁿ by definition. (By the previous point applied to f⁻¹_B the basis B is orthonormal if and only if f_B is an isomorhism of inner product spaces, and the equation used as definition states just that.)

So up to isomorphism Rⁿ with the standard inner product is the unique n-dimensional inner product space.

For two inner product spaces V,W of dimension n, there are as many isomorphisms of inner product spaces as there are orthonomal bases in W (or in V); the set is also in bijection with the set of automorphsims of the standard inner product space Rn, which is the orthogonal group On(R).

On a given space V there are as many different (though isomorphic) inner products as there are cosets in On(R)∖GLn(R). (Both the sets of isomorphisms and of different inner products are infinite in general, so "as many" should be interpreted as "naturally in bijection with".) Explanation for the latter correspondence: fixing B, the cooordinate map V→Rn corresponding to any basis of V is of the form g∘fB for some g∈GLn(R), which defines an inner product on V, and g1,g2∈GLn(R) define the same inner product iff the standard basis of Rn transported to V by (g1∘fB)−1 and then back to Rn by g1∘fB give an orthonormal basis, which means g2∘g−11∈On(R) of g2∈On(R)g1. The set of inner products on Rn is also in bijection with the set of positive definite symmetric n×n matrices.

Links

1. math.stackexchange
   1. isomorphisms-of-inner-product-spaces
   2. orthonormal-basis-isomorphism-preserving-dot-product
   3. isomorphic-as-inner-product-spaces
   4. isomorphism-of-inner-product-spaces

Talk

Diario ottobre2019

ho ripreso a pensare al prodotto interno.

‖ x + y ‖² + ‖ x − y ‖² = 2 ‖ x ‖² + 2 ‖ y ‖²