Product of Banach Spaces as Banach Spaces: Norms and Completeness

A Banach space is a complete normed vector space, a fundamental structure in functional analysis. A natural question arises: Under what conditions, if any, is the product of two Banach spaces also a Banach space? This article will explore the conditions and norms required for the product of two Banach spaces to be a Banach space.

Introduction to Banach Spaces

Let us first briefly recap what a Banach space is. A Banach space is a vector space V over the field of real or complex numbers, equipped with a norm that satisfies certain properties, and is also a complete metric space with respect to the distance induced by the norm. In simpler terms, a Banach space is a space where all Cauchy sequences converge.

Product of Two Banach Spaces

Assume that we have two Banach spaces X and Y. The product space X × Y consists of all pairs (x, y) where x ∈ X and y ∈ Y. This product space can be studied as a vector space over the same field as X and Y, by defining the operations component-wise. That is, for (x?, y?), (x?, y?) ∈ X × Y and scalars λ and μ, we have:

(x?, y?) (x?, y?) (x? x?, y? y?) λ(x?, y?) (λx?, λy?)

We now consider the problem of equipping X × Y with a norm. One natural candidate is the multiplicative norm. This norm is defined as:

xy_{X×Y} x_X y_Y

where x_X and y_Y denote the norms of x in X and y in Y, respectively.

To understand whether XY_{X×Y} forms a Banach space with respect to this norm, we need to verify that the normed space (X × Y, XY_{X×Y}) is complete. This means that every Cauchy sequence in X × Y converges to a limit also in X × Y.

Completeness of the Product Space with Multiplicative Norm

Let {(x?, y?)} be a Cauchy sequence in X × Y with the multiplicative norm. This means that for every ε > 0, there exists N such that for all m, n ≥ N, we have:

xy_{X×Y} x?,y? - x?,y?

Since X and Y are Banach spaces, it follows that the sequences {x?} and {y?} are both Cauchy in X and Y, respectively. Because X and Y are complete, there exist x ∈ X and y ∈ Y such that x? converges to x and y? converges to y in the respective norms.

It remains to show that (x?, y?) converges to (x, y) in the product norm XY_{X×Y}. We need to show that:

(x?, y?) - (x, y)_{X×Y} x? - x_X y? - y_Y → 0 as n → ∞

Since x? → x and y? → y in their respective norms, we can find N such that for all n ≥ N, we have:

x? - x_X

y? - y_Y

Thus, for n ≥ N:

x? - x_X y? - y_Y ≤ x? - x_X x_X (y? - y_Y) ≤ x? - x_X x_X y? - y_Y

This shows that (x?, y?) converges to (x, y) in the product norm. Therefore, (X × Y, XY_{X×Y}) is a Banach space.

Alternative Norms on the Product Space

While the multiplicative norm is a natural choice, it is not the only one. Another common norm for the product space X × Y is the maximum norm defined as:

((x, y))_{X × Y} max(x_X, y_Y)

For this norm, we need to check if the space (X × Y, ((x, y))_{X × Y}) is complete. Indeed, if we have a Cauchy sequence (x?, y?) in this norm, the sequences {x?} and {y?} are Cauchy in X and Y, respectively. Since X and Y are Banach, they converge to some x ∈ X and y ∈ Y. Furthermore, (x?, y?) converges to (x, y) in the maximum norm. Therefore, (X × Y, ((x, y))_{X × Y}) is also a Banach space.

Conclusion

In conclusion, the product of two Banach spaces X and Y can indeed be a Banach space under the appropriate norm. This includes the multiplicative norm and the maximum norm. Understanding these norms and the conditions for completeness is crucial in various applications of functional analysis.

Keywords

Banach space, Norm, Completeness

Further Reading

For more information on Banach spaces and related topics, consider exploring the following resources:

Banach Space on Wikipedia Introduction to Banach Spaces by Renzo L. Ricca Course Notes on Banach Spaces