Can we define a norm on every vector space?

Can we define a norm on every vector space?

7 Answers. Every (real or complex) vector space admits a norm. Indeed, every vector space has a basis you can consider the corresponding «ℓ1» norm.

Does every finite dimensional vector space have a norm?

Try books on the topic of “topological vector spaces”: It is a theorem that every finite dimensional real or complex vector space has a norm, and that all norms are equivalent. Correspondingly, there are infinite dimensional topological vector spaces that don’t have a norm that induces the topology.

Is it always possible to define a norm on each vector space over R or C?

Yes. Recall that every(finite or infinite dimensional) vector space has an algebraic/Hamel basis using axiom of choice. Write any vector in terms of this basis, and take the maximum coordinate. It can be verified that this defines a norm on the space.

Is L2 a vector space?

Again, one can prove that L2 is a vector space; that is, it is closed under addition and scalar multiplication. Unlike finite-dimensional spaces, such inequalities do not hold between any pair of norms.

Which of the following is not vector space?

A vector space needs to contain 0⃗ 0→. Similarily, a vector space needs to allow any scalar multiplication, including negative scalings, so the first quadrant of the plane (even including the coordinate axes and the origin) is not a vector space.

What is the norm of a vector?

The length of the vector is referred to as the vector norm or the vector’s magnitude. The length of a vector is a nonnegative number that describes the extent of the vector in space, and is sometimes referred to as the vector’s magnitude or the norm.

Is there a vector space without a norm?

$\\begingroup$Yes, there are vector spaces without a norm. Yes, there are normed spaces that are not complete in the norm of the space. Some asides: there are normed spaces for which the norm is not induced by any inner product. There are normed spaces which are complete in the norm induced by the inner product.

What is a non-metrizable vector space?

A non-metrizable vector space is a topological vector space whose topology does not arise from any metric. These are rather common in functional analysis. For example, if $X$ is a Banach space, then the weak-* topology on $X^*$ is never metrizable unless $X$ is finite-dimensional.

Can every vector space be equipped with an inner product?

Christian Blatter’s answer shows that, assuming the axiom of choice, every vector space can be equipped with an inner product. Without the axiom of choice, this can fail. As I show in Inner product on C (R), it is consistent with ZF+DC that the vector space C (R) of continuous functions on R does not admit any inner product, nor even any norm.