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I'm confused at the construction of the Riesz Representation for a Hilbert space.

Given a Hilbert space $H$, we define $J(y) := ( * \space | \space y) \in H'$

This is an element of the dual space $H'$. My confusion that I want to clarify is this: on the LHS of the inner product, do we just fix an $x \in H$ and vary the $y$? If so, 'which' $x$ do we fix? Or does it not matter, as long as we fix one?

I know that $J: H \rightarrow H' $ is supposed to be bijective, and we identify $y \in H$ with $J(y) \in H'$, hence my confusion about what element to fix on the LHS.

Thanks so much and apologies if this is a really basic question.

JJJ
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  • Actually, I think I have got this wrong, for each $y \in H$, we have an element of the dual space $J(y) \in H'$. So we fix $y$ and vary the LHS. Would that be correct? – JJJ Jun 18 '16 at 11:35

1 Answers1

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The open argument in the inner product is the argument of the linear funcitonal that is defined in this way. I.e. $J(y)$ is an element of $H^{\prime}$, so it is a linear functional $H\colon \to \mathbb{R}$. This means that it is a function that takes as argument elements of $H$. So you have $J(y)(x)=(x\mid y)$.

I find it easier to think about this bijection the other way around, namely if you have $f\in H^{\prime}$ then there exists some $y\in H$ such that $f(x)=(x\mid y)$ for all $x\in H$.

Nikolai
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