Intereting Posts

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Is every symmetric bilinear form on a Hilbert space a weighted inner product?

i.e. can I write that $b(u,v) = (wu,v)_H$ for all $u, v \in H$?

I am not sure about this. Maybe something to do with Riesz theorem..

- What is the norm of the operator $L((x_n)) \equiv \sum_{n=1}^\infty \frac{x_n}{\sqrt{n(n+1)}}$ on $\ell_2$?
- Inequivalent Hilbert norms on given vector space
- Closed subspace $M=(M^{\perp})^{\perp}$ in PRE hilbert spaces.
- Counterexample for the stability of orthogonal projections
- Finding the min of an integral
- (From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$
- If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$
- proof for a basis in $L^2$
- Bounded linear operator on a Hilbert space
- Orthogonal complement of a Hilbert Space

Since you’re interested in bilinear forms, I’ll assume you’re working with a real Hilbert space $H$; in the complex case, replace “bilinear” with “sesquilinear.”

- First, suppose that you have a
*bounded*bilinear form $b$, so that there exists some constant $C \geq 0$ such that $b(u,v) \leq C\|u\|\|v\|$ for all $u$, $v \in H$. Then for all $u \in H$, $b(u,\cdot)$ defines a bounded linear functional on $H$ (why?), so that by the Riesz theorem, $b(u,\cdot) = (w(u),\cdot)_H$ for a unique $w(u) \in H$; check (how?) that $u \mapsto w(u)$ defines a bounded linear operator on $H$. - Now, suppose in addition that $b$ is symmetric. Check that $w$ is therefore self-adjoint. In fact, $b$ is symmetric if and only if $w$ is self-adjoint (why?).
- Finally, suppose in addition that $b$ is positive-definite, i.e., $b(u,u) > 0$ for all $u \neq 0$. Check, then, that $w$ is therefore positive-definite, so that $b$ is indeed a weighted inner product. In fact, $b$ is positive-definite (and thus a weighted inner product) if and only if $w$ is positive definite (why?).

Throughout all this, all you really need is to remember what the Riesz theorem says, and what it means for an operator to be self-adjoint and positive-definite. In particular, observe the role (indeed, necessity and sufficiency) of the hypotheses of boundedness, symmetry (which you already had), and positive-definiteness.

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