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196 ALGEBRA

Properties of an orthonormal basis of a unitary space:
1. Let i 1 , ... , i n be an orthonormal basis in a unitary space. Then the scalar product of
two elements x = x 1 i 1 + ··· + x n i n and y = y 1 i 1 + ··· + y n i n is equal to the sum
x ⋅ y = x 1 ¯y 1 + ··· + x n ¯y n .

2. The coordinates of any vector in an orthonormal basis i 1 , ... , i n are equal to the scalar
products of this vector and the vectors of the bases (or the projections of this element
on the axes in the direction of the corresponding basis vectors):

x k = x ⋅ i k (k = 1, 2, ... , n).

Two unitary spaces V and V are said to be isomorphic if there is a one-to-one corre-
2
spondence between their elements satisfying the following conditions: if elements x and
y of V correspond to elements 2 x and 2 y of V,then x + y corresponds to 2 x + 2 y; the element
2
λx corresponds to λ2 x for any complex λ; the scalar product (x ⋅ y) V is equal to the scalar
product (2 x ⋅ 2 y) .
2 V
THEOREM. Any two n-dimensional unitary spaces V and V are isomorphic.
2

5.4.3. Banach Spaces and Hilbert Spaces

5.4.3-1. Convergence in unitary spaces. Banach space.
Any normed linear space is a metric space with the metric (5.4.2.2).
A sequence {b s } of elements of a normed space V is said to be convergent to an element
b V as s →∞ if lim b s – b = 0.
s→∞
Aseries x 0 +x 1 +··· with terms in a normed space is said to be convergent to an element
n ∞
x (called its sum; one writes x = lim x k = x k ) if the sequence of its partial sums
n→∞ k=0 k=0
n
3 3
forms a sequence convergent to x, i.e., lim 3x – x k3 = 0.
3
3
k=0
n→∞
A normed linear space V is said to be complete if any sequence of its elements s 0 ,
s 1 , ... satisfying the condition
lim s n – s m = 0
n,m→∞
is convergent to some element s of the space V.
A complete normed linear space is called a Banach space.
Remark. Any finite-dimensional normed linear space is complete.

5.4.3-2. Hilbert space.
A complete unitary space is called a Hilbert space.
Any complete subspace of a Hilbert space is itself a Hilbert space.
PROJECTION THEOREM. Let V 1 be a complete subspace of a unitary space V. Then for
any x V, there is a unique vector x p V 1 such that

min x – y = x – x p .
y V 1