Final answer:
The inequality ∣∣u∣∣_{∞} ≤ ∣∣u∣∣_{1} indicates that in any n-dimensional real vector space, the infinity norm of a vector is always less than or equal to its 1-norm. This reflects the fact that the sum of all components' absolute values will always be greater than or equal to the greatest component's absolute value.
Step-by-step explanation:
The inequality ∣∣u∣∣_{∞} ≤ ∣∣u∣∣_{1} pertains to the relationship between two different norms on Rⁿ, specifically the infinity norm (also known as the maximum norm) and the 1-norm (or taxicab norm / Manhattan norm).
In a vector u in n-dimensional real space, the infinity norm is the absolute value of the component of u with the largest magnitude, whereas the 1-norm is the sum of the absolute values of all components of u. The inequality states that the infinity norm of any vector u will always be less than or equal to its 1-norm.
The given inequality can be understood by noting that the largest absolute value (that defines the infinity norm) can never exceed the sum of all absolute values (which defines the 1-norm). This is because when you add positive quantities (such as absolute values), the total is always greater than or equal to each individual quantity.