Question

Let \(X\) be any normed linear space and \(X'\) be the dual space of \(X\). Which of the following statements is/are TRUE?

• A. If \(X\) is separable and \(X'\) is non-separable, then \(X\) must be reflexive.
• B. If \(X\) is separable and \(X'\) is non-separable, then \(X\) cannot be reflexive.
• C. If \(X\) is reflexive, then \(X'\) is reflexive.
• D. If \(X'\) is separable, then \(X\) is separable.

Hint:

Use the theorem that a separable dual forces the space to be separable, plus the fact that reflexivity of a space and its dual always go together, then argue by contrapositive.

Answer 1

The Correct Option is B, C, D

Step 1: Recall the basic theorem linking separability of a space and its dual.
A standard result in functional analysis says: if the dual space \(X'\) of a normed linear space \(X\) is separable (has a countable dense subset), then \(X\) itself is separable.
This gives statement (D) directly: "If \(X'\) is separable, then \(X\) is separable" is TRUE.
The converse of this fact does not hold in general, a separable space can have a non-separable dual, the classic example being \(X = \ell^1\), which is separable, while its dual \(X' = \ell^\infty\) is not separable.

Step 2: Recall the reflexivity duality theorem.
Another standard theorem says a normed linear space \(X\) is reflexive if and only if its dual \(X'\) is reflexive.
So if \(X\) is reflexive, its dual \(X'\) must also be reflexive. This gives statement (C): "If \(X\) is reflexive, then \(X'\) is reflexive" is TRUE.

Step 3: Use these two facts together to test statements (A) and (B).
Suppose \(X\) is separable and, for the sake of argument, that \(X\) is reflexive.
Since \(X\) is reflexive, \(X''\) (the dual of \(X'\)) is isometrically the same as \(X\), which is separable, so \(X''\) is separable too.
Applying the Step 1 theorem to the space \(X'\) in place of \(X\): since the dual of \(X'\), namely \(X''\), is separable, \(X'\) itself must be separable.
So if \(X\) is separable and reflexive, then \(X'\) must be separable. Taking the contrapositive: if \(X\) is separable and \(X'\) is NOT separable, then \(X\) cannot be reflexive.
This matches statement (B), so (B) is TRUE, and it directly rules out statement (A), which claims the opposite ("X must be reflexive"), so (A) is FALSE.
The classical example \(X=\ell^1\) (separable) with \(X' = \ell^\infty\) (not separable) confirms this: \(\ell^1\) is well known to NOT be reflexive.

Step 4: Final answer.
Statements (B), (C), and (D) are TRUE, and statement (A) is FALSE.
\[ \boxed{\text{(B), (C) and (D) are TRUE}} \]