## Dual spaces

Suppose \$latex {(A,||cdot||_A)}&fg=000000\$ and \$latex {(B,||cdot||_B)}&fg=000000\$ are Banach space, \$latex {A^*}&fg=000000\$ and \$latex {B^*}&fg=000000\$ are their dual spaces. If \$latex {Asubset B}&fg=000000\$ with \$latex {||cdot||_Bleq C||cdot||_A}&fg=000000\$, then

\$latex displaystyle i:Amapsto B&fg=000000\$

\$latex displaystyle quad xrightarrow x&fg=000000\$

is an embedding. Let us consider the relation of two dual spaces. For any \$latex {fin B^*}&fg=000000\$

\$latex displaystyle |langle f,xrangle|=|f(x)|leq ||f||_{B^*}||x||_Bleq C||f||_{B^*}||x||_Aquad forall, xin A&fg=000000\$

Then \$latex {f|_{A}}&fg=000000\$ will be a bounded linear functional on \$latex {A}&fg=000000\$

\$latex displaystyle i^*:B^*mapsto A^*&fg=000000\$

\$latex displaystyle qquad frightarrow f|_A&fg=000000\$

is a bounded linear operator.

In a very special case that \$latex {A}&fg=000000\$ is a closed subset of \$latex {B}&fg=000000\$ under the norm \$latex {||cdot||_B}&fg=000000\$, one can prove \$latex {i^*}&fg=000000\$ is surjective. In fact \$latex {forall,gin A^*}&fg=000000\$ can be extended to \$latex {bar{g}}&fg=000000\$ on \$latex {B}&fg=000000\$ by Hahn-Banach thm such that \$latex {i^*bar{g}=g}&fg=000000\$. Then

\$latex displaystyle A^*=B^*/ker i^*.&fg=000000\$

Let us take \$latex {A=H^1_0(Omega)}&fg=000000\$ and \$latex displaystyle B=H^1(Omega)&fg=000000\$…

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