Spaces and Convergences


Metric spaces: a set where a notion of distance (called a metric) between elements of the set is defined.

Vector spaces or Linear space: a mathematical structure formed by a collection of vectors.

Normed vector spaces: a Vector space with a norm defined. The norm is an abstraction of our usual concept of length.

Any normed vector space is a metric space by defining d(x,y)=||y-x||.

Metric space may not be a vector space.

Inner product spaces: Vector spaces endowed with inner product.

An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space.

A normed vector spaces is also an inner product spaces if and only if its norm satisfies the parallelogram law.

Banach spaces: complete normed vector spaces.

Hilbert spaces: complete inner product spaces.

Weak convergence(in normed vector space)

Suppose X is a normed vector space, X^{*} is the continuous dual of X , and x_{1},x_{2},\cdots is a sequence in X, x_{o}\in X. Then we say that {x_{n}} converges weakly to x_{0} if \lim_{n\to\infty}f(x_{n})=f(x_{0}) for every f\in X^{*}. The notation for this is x_{n}\stackrel{w}{\to}x_{0}.

Strong convergence(in normed vector space)

||x_{n}-x_{0}||\to0 (n\to\infty)

The notation for this is x_{n}\to x_{0} or x_{n}\stackrel{s}{\to}x_{0}. We also call it convergence in norm.

Weak* convergence

Suppose X is a normed vector space, \{f_{n}\}\subset X^{*}, f_{0}\subset X^{*}(n=1,2,\cdots).

It is the same as strong convergence for a series of operators.

For any x\in X, f_{n}(x)\to f_{0}(x). The notation for this is f_{n}\stackrel{w*}{\to}f_{0}.

Strong convergence

If \{f_{n}\} converse to f_{0} by the norm on X^{*}, or ||f_{n}-f_{0}||\to0(n\to\infty)

It is the same as uniformly convergence for a series of operators.

References:

1. http://en.wikipedia.org

2. Optimization by Vector Space Methods, Luenberger, David G., 1998

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