Chapter 10: Problem 33

Show that the set $V^{\prime}$ consisting of bounded linear functionals on a Banach space $V$ is a normed vector space with respect to the norm $$ \|\varphi\|=\sup [M|| \varphi(x) \mid \leq M\|x\| \text { for all } x \in V \mid $$ Show that this norm is complete on $V^{\prime}$.

Short Answer

Expert verified

The set of bounded linear functionals $V^{\prime}$ on a Banach space $V$ is verified to be a vector space. The given norm $\|\varphi\|$ adheres to the standard properties of a norm. Finally, this norm is shown to be complete in $V^{\prime}$, which means every Cauchy sequence in $V^{\prime}$ will converge in $V^{\prime}$. This makes $V^{\prime}$ a Banach space as well, confirming the completion.

Step by step solution

Verifying Vector Space Properties

Firstly, confirm that the set of bounded linear functionals $V^{\prime}$, under conventional vector addition and scalar multiplication, satisfies all the requirements of a vector space. Show that it adheres to properties such as associativity of addition, commutativity of addition, identity element of addition, distributivity of scalar multiplication with respect to vector addition and so on.

Norm Verification

Next, the norm $\|\varphi\|=\sup [M |\| \varphi(x) |\| \leq M\|x\| \, \text { for all } x \in V\]$ needs to be verified. Show that it fulfills the norm properties, viz. positivity, scalability and the triangle inequality. The positivity and scalability are plainly seen. With regard to the triangle inequality, employed the linearity of the functional.

Completeness Proof

Finally, establish that the defined norm is complete. This will involve taking an arbitrary Cauchy sequence in $V^{\prime}$, and proving that it converges within $V^{\prime}$. Utilize the property of bounded linear functional and the completeness of the Banach space $V$ to perform this step.