The L2-norm (||·||2) of any vector 𝑥 is bounded by its L1-norm (||·||1):
- ||𝑥||2 ≤ ||𝑥||1
This fact generalizes to L𝑝-norms in that the L𝑝-norm (||·||𝑝) of any given vector 𝑥 does not grow with 𝑝:
- ||𝑥||𝑝+𝑎 ≤ ||𝑥||𝑝
for any vector 𝑥 and real numbers 𝑝≥1 and 𝑎≥0. (In fact, this remains true for 0<𝑝<1 and 𝑎≥0)