Normed and topological vector spaces

In summary, topological vector spaces are more general than normed ones because they allow for a wider range of topologies, including those that cannot be induced by a norm. The standard way to define a topology for a normed space is through open balls, but there are examples, such as the set of all sequences of real numbers, where this topology cannot be derived from a norm. On the other hand, there are topological vector spaces that cannot be derived from any metric or norm, such as the space with the trivial topology.
  • #1
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Can someone explain how topological vector spaces are more general than normed ones. So this means that if I have a normed vector space, [tex]X[/tex], it would have to induce a topology. I was thinking the base for this topology would consist of sets of the form [tex]\{y\in X: ||x-y||<\epsilon, \textrm{for some x$x\in X$ and $\epsilon>0$}\}[/tex], which is the analogue of the open ball centred at y with radius epsilon. Is this correct?
 
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  • #2
Yes, given a normed space, your definition is the standard way to define a topology, and the result is a topological vector space.

Now consider the set [tex]\mathbb{R}^\mathbb{N}[/tex] of all sequences of real numbers. With the obvious vector operations and the topology of pointwise convergence this becomes a topological vector space. But it can be shown that there is no norm for this space that produces this topology.
 
  • #3
On the other hand, consider any vector space, V, with the topology in which the only open sets are V itself and the empty set. This is a topological vector space which cannot be derived from any metric, much less a norm.
 

Related to Normed and topological vector spaces

1. What is a normed vector space?

A normed vector space is a mathematical concept that combines the properties of a vector space and a norm. A vector space is a set of objects that can be added together and multiplied by scalars, while a norm is a function that assigns a non-negative length to each element in the vector space. In a normed vector space, the norm function is defined and satisfies certain properties, such as the triangle inequality and positive homogeneity.

2. How is a normed vector space different from a topological vector space?

A topological vector space is a vector space that is equipped with a topology, which is a set of rules for determining which sequences of points converge. In contrast, a normed vector space is a special type of topological vector space where the topology is induced by the norm function. This means that the topology of a normed vector space is determined by the norm, while the topology of a topological vector space can be more general.

3. What is the role of convergence in normed and topological vector spaces?

Convergence is an important concept in both normed and topological vector spaces. In a normed vector space, a sequence of points converges if the distance between the points approaches 0 as the sequence goes to infinity. In a topological vector space, the topology determines which sequences converge. This allows for the study of continuity and convergence of functions in these spaces.

4. How are normed and topological vector spaces used in applications?

Normed and topological vector spaces are used in a variety of applications in mathematics, physics, and engineering. For example, they are used in functional analysis to study the properties of function spaces. They are also used in optimization problems, such as finding the shortest path between two points. In physics, they are used to model physical systems and study their properties.

5. What are some examples of normed and topological vector spaces?

Some common examples of normed and topological vector spaces include Euclidean spaces, function spaces, and sequence spaces. Euclidean spaces, such as R^n, are normed vector spaces with the usual Euclidean norm. Function spaces, such as L^p spaces, are normed vector spaces where the norm is based on the integrals of functions. Sequence spaces, such as l^p spaces, are normed vector spaces where the norm is based on the absolute values of the terms in a sequence.

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