Let $X$ be a Hilbert space.
$J:X\rightarrow X',\hspace{1cm}J(x):=(\cdot,x)$
is a complex conjugated isometric isomorphism between $X$ and it's dual space $X'$.
Would there be any problems as a result of changing the definition to $J(x):=(x,\cdot)$ ?
2026-03-25 08:12:36.1774426356
Riesz representation theorem: Does the order matter?
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In the case of complex scalars $J(x)=(x,.)$ would not define a linear map on $X$ for fixed $x$. It would be conjugate linear.