Question

show that a norm on a vector space X is a sublinear functional on X

Answer 1

Answer on Question #70186 – Math – Functional Analysis

Question

Show that a norm on a vector space $X$ is a sublinear functional on $X$.

Solution

Let's $f(x) = ||x||$. Obviously, $f(x)$ is a function from a vector space $X$ to the scalar field $\mathbb{R}$.

1. $\forall x\in X,\quad \forall a\in \mathbb{R}_{+},\quad f(x) = ||a\cdot x|| = |a|\cdot ||x|| = a\cdot ||x|| = a\cdot f(x),\quad \text{due to the multiplicative property of a norm.}$

2. $\forall x, y \in X, f(x + y) = ||x + y|| \leq ||x|| + ||y|| = f(x) + f(y),$ because of the triangle inequality.

Correctness of the statement and both properties (positive homogeneity and subadditivity) were proved.

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