Banach Space

A Banach space is a normed space that is complete in the metric induced by its norm. Two norms and are called equivalent if they give the same topology, which is equivalent to the existence of constants such that

(1)

and

(2)

hold for all . These bounds show that equivalent norms have the same Cauchy sequences and preserve completeness.

In the finite-dimensional case, all norms are equivalent. An infinite-dimensional space can have many different norms.

A basic example is -dimensional Euclidean space with the Euclidean norm. More generally, every finite-dimensional normed space over or is a Banach space. Infinite-dimensional examples include vector spaces of functions. For example, the set of continuous functions on a closed interval with , equipped with the norm

(3)

is a Banach space, where denotes the supremum.

On the other hand, the set of continuous functions on the unit interval with the norm of a function given by

(4)

is not a Banach space because it is not complete. For instance, for integers , the Cauchy sequence of functions

(5)

converges in the integral norm to the step function that is 1 on and 0 on . The error is , so the triangle inequality makes a Cauchy sequence. Any continuous function agreeing with almost everywhere would be identically 1 on and identically 0 on , contradicting continuity at (Axler 2026, p. 165).

Hilbert spaces with their norm given by the inner product are examples of Banach spaces. While a Hilbert space is always a Banach space, the converse need not hold. Therefore, it is possible for a Banach space not to have a norm given by an inner product. For example, on the space of continuous functions on , the supremum norm fails the parallelogram law required by an inner product (Axler 2026, p. 220). For and , all four norms , , , and are 1, so

(6)

Renteln and Dundes (2005) give the following (bad) mathematical joke about Banach spaces:

Q: What's yellow, linear, normed, and complete? A: A Bananach space.

The Banach-Mazur distance quantifies the least distortion between finite-dimensional normed spaces under an invertible linear transformation. For origin-symmetric convex bodies, it equivalently compares containment between a linear image and a dilate of that image.