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.

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