**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 .

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, is the continuous dual of , and is a sequence in , . Then we say that {} converges weakly to if for every . The notation for this is .

**Strong convergence**(in normed vector space)

()

The notation for this is or . We also call it convergence in norm.

**Weak* convergence**

Suppose X is a normed vector space, , ().

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

For any , . The notation for this is .

**Strong convergence**

If converse to by the norm on , or ()

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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