Vector spaces
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Ring
아래와 같은 2개의 연산을 만족하는 집합
(R,+) is a commutative group, i.e.,
associative addition
commutative addition
additive zero and inverse
(R,*) is a semi-group, i.e., a*(b*c) = (a*b)*c
Vector space
Linear vector space X consist of
- a set of elements, called vectors
- a set of scalars in a field F
- two operators, called vector addition and scalar multiplication
Normed Vector space
Normed vector spaces with a real-valued function, || ||, such as
Nonnegativity
Strict positivity
Homogeneity
Triangle inequality
Banach space
A complete normed vector space
어떤 space 내부 혹은 경계에 빠진 점이 없는 거리 공간
A normed vector space X is complete if every Cauchy sequence from X has a limit in X
| A sequence {xn} in a normed space is said to be a Cauchy sequence if | xn - nm | -> 0 as n,m -> infty; i.e., given epsilon>0, there is an integer N such that | xn - xm | <epsilon for n,m >N |
Hilbert space
A Banach space with an inner product < , >
Symmetry
Additivity
Homogeneity
Strict positivity
각 space 의 중요성 및 쓰임에 대해…