If $(U,〈\;⋅\;,\;⋅\;〉),H$ are Hilbert spaces, $W\in U$, $Y\in H$, $Z\in L(U, H)$ and $f\in L(H,L(H,\mathbb R))$, then $〈Y,fZW〉=〈ZW,fY〉$

Question

Let

• $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be Hilbert spaces
• $W\in U$, $Y\in H$ and $Z\in\mathfrak L(U, H)$$^1$
• $f\in\mathfrak L\left(H,\mathfrak L\left(H,\mathbb R\right)\right)$

How can we show that $$\langle Y,fZW\rangle=\langle ZW,fY\rangle\;?$$

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Let $\mathfrak L(A,B)$ be the space of bounded and linear operators from $A$ to $B$.

Answer 1

This seems false. Take $U=H$, $Z=Id$, and identify $L(H,\mathbb{R})$ with $H$. Take and orthogonal base $e_1,e_2\ldots,$ and define $f(e_1)=e_2$, $f(e_i)=0$ for $i>1$, extend by linearity. Then $$ \langle e_2,f(e_1)\rangle=1\neq0\langle e_1,f(e_2)\rangle. $$