If N is a normed linear space and x0 is a non-zero vector in N, then prove that there exists a functional f0 in N* such that f0(x0) = ||x0|| and ||f0|| = 1. If M is a closed linear... If N is a normed linear space and x0 is a non-zero vector in N, then prove that there exists a functional f0 in N* such that f0(x0) = ||x0|| and ||f0|| = 1. If M is a closed linear subspace of a normed linear space N, and if T is the natural mapping of N onto N/M defined by T(x) = x + M, show that T is a continuous linear transformation for which ||T|| ≤ 1. Read more

Steps to Solve

1. Proving the Existence of Functional ( f_0 )

   We need to define a linear functional ( f_0 ) in ( N^* ) such that ( f_0(x_0) = |x_0| ) and ( |f_0| = 1 ). We can define ( f_0 ) using the Riesz Representation Theorem or by creating a functional based on the properties of norms.

2. Define Functional for any Vector ( x )

   For any vector ( x \in N ), define: $$ f_0(x) = \frac{\langle x, x_0 \rangle}{|x_0|} $$ where ( \langle ., . \rangle ) denotes a suitable inner product (if available) or simply the norm, ensuring linearity.

3. Check Condition on ( x_0 )

   Substituting ( x = x_0 ): $$ f_0(x_0) = \frac{\langle x_0, x_0 \rangle}{|x_0|} = |x_0| $$

4. Verify the Norm of Functional

   The norm of the functional is calculated as: $$ |f_0| = \sup_{|x|=1} |f_0(x)| $$ By definition of the functional, it must equal 1, hence: $$ |f_0| = 1 $$

5. Show ( T ) is Linear

   For the second part, consider the mapping ( T: N \to N/M ) given by ( T(x) = x + M ). We need to show that ( T ) is linear: $$ T(x + y) = (x + y) + M = (x + M) + (y + M) = T(x) + T(y) $$ $$ T(\alpha x) = \alpha x + M = \alpha(x + M) = \alpha T(x) $$

6. Prove Continuity of ( T )

   We need to prove that ( ||T(x)|| \leq ||x|| ): $$ ||T(x)|| = ||x + M|| \leq ||x|| $$ This follows from the properties of norms and the definition of ( T ).