how can i proove that every vector norm on C subscript n is continuos in the vector components the set of complex numbers
To prove that every vector norm on C subscript n is continuous in the vector components, we can follow these steps:
Step 1: Choose any vector norm on C subscript n. Let's denote it as || . ||.
Step 2: Recall that a vector norm satisfies three properties: positivity, scalar multiplication preservation, and triangle inequality.
Step 3: Let's focus on proving the continuity of the norm with respect to the vector components.
Step 4: Consider a vector x = (x1, x2, ..., xn) in C subscript n.
Step 5: Recall that the norm of a vector x is defined as ||x|| = sqrt(|x1|^2 + |x2|^2 + ... + |xn|^2), where |xi| represents the modulus of the complex number xi.
Step 6: Now, let's fix one component of the vector, say x1, and vary the other components (x2, x3, ..., xn). We define a function f(x1) = sqrt(|x1|^2 + |x2|^2 + ... + |xn|^2).
Step 7: Since the modulus operator | | is continuous for complex numbers, each term |xi|^2 in the function f(x1) is continuous.
Step 8: We can use the properties of continuity to conclude that the sum of continuous functions is also continuous.
Step 9: Finally, since the square root function is continuous, the overall function f(x1) = sqrt(|x1|^2 + |x2|^2 + ... + |xn|^2) is continuous.
Step 10: By applying the same reasoning to the other vector components, we can show that every component of the vector norm is continuous.
Thus, we have proved that every vector norm on C subscript n is continuous in the vector components.