Evans p298 - Lax Milgram Theorem

Let $H$ be a Hilbert space. Denote the inner product by $(\cdot,\cdot)$ and the natural dual pairing of spaces by $\langle\cdot,\cdot\rangle$.

He gives us bilinear form $B[\cdot,\cdot]:H\times H \to \Bbb R$ and then he says that for each fixed element $u\in H$ the mapping $v\mapsto B[u,v]$ is a bounded linear functional on $H$, and hence Riesz representation theorem gives us a unique element $w\in H$ such that: $$B[u,v]=(w,v),\qquad (v\in H)$$

How exactly does this follow from RRT? Riesz gives us that any time we consider $u^*\in H^*$ there is some $u\in H$ such that: $$\langle u^*,v\rangle = (u,v).$$

So I don't follow how the RRT gives the above, I mean unless the RRT doesn't hold just for the natural pairing, but rather for any bilinear form?


Well, $v \to B[u,v]$ is a linear functional on $H$. The RRT tells you that any bounded linear functional $l$ on $H$ can be represented as $l(v)=(u,v)$ for some $v$, and that the map $l \to u$ is in fact an isometric isomorphism. That is why you usually write $l=u^*$. Hence, since $v \to B[u,v]$ is a bounded linear functional you will find a $w$ such that $B[u,v]=(w,v)$, you may write $B[u,\cdot]=w^*$ if you like.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.