Question

Use principle of uniform boundedness to show that a set E in a normed space X is bounded if f(E) bounded in for K for every f ∈ X'

Answer 1

To prove that a set E in a normed space X is bounded, we can use the principle of uniform boundedness. If f(E) is bounded in K for every f in X', then E must also be bounded in X.

Step-by-step explanation:

To prove that a set E in a normed space X is bounded, we can use the principle of uniform boundedness. If f(E) is bounded in K for every f in X', then E must also be bounded in X.

1. Assume that f(E) is bounded for every f in X'.
2. Let x be an element in E. Then, for every f in X', f(x) is in f(E), which is bounded in K.
3. Let M be the supremum of all |f(x)| for all f in X'. Then, for every x in E, we have |f(x)| <= M for all f in X'.

Therefore, the set E is bounded in X.