Riesz representation theorem of bounded sesquilinear forms
Bounded sesquilinear forms
Let ${H}_{1}$, ${H}_{2}$ be two Hilbert spaces^{}.
Definition  A sesquilinear form^{} $[\cdot ,\cdot ]:{H}_{1}\times {H}_{2}\to \u2102$ is said to be bounded^{} if there is a constant $C\ge 0$ such that
$[\xi ,\eta ]\le C\parallel \xi \parallel \parallel \eta \parallel $ 
for all $\xi \in {H}_{1}$ and $\eta \in {H}_{2}$.
Bounded sesquilinear forms are precisely those which are continuous from ${H}_{1}\times {H}_{2}$ to $\u2102$.
Examples :

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When ${H}_{1}$ and ${H}_{2}$ are the same Hilbert space, denoted by $H$, the inner product^{} $\u27e8\cdot ,\cdot \u27e9$ in $H$ is itself a bounded sesquilinear form. The boundedness condition follows from the CauchySchwarz inequality.

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Let $T:{H}_{1}\to {H}_{2}$ be a bounded linear operator and denote by $\u27e8\cdot ,\cdot \u27e9$ the inner product in ${H}_{2}$. The function $[\cdot ,\cdot ]:{H}_{1}\times {H}_{2}\to \u2102$ defined by
$[\xi ,\eta ]:=\u27e8T\xi ,\eta \u27e9$ is a bounded sesquilinear form. The boundedness condition follows from the CauchySchwarz inequality and the fact that $T$ is bounded.
Riesz representation of bounded sesquilinear forms
The second example above is in fact the general case. To every bounded sesquilinear form one can associate to it a unique bounded operator^{}. That is content of the following result:
Theorem  Riesz  Let ${H}_{1}$, ${H}_{2}$ be two Hilbert spaces and denote by $\u27e8\cdot ,\cdot \u27e9$ the inner product in ${H}_{2}$. For every bounded sesquilinear form $[\cdot ,\cdot ]:{H}_{1}\times {H}_{2}\to \u2102$ there is a unique bounded linear operator $T:{H}_{1}\to {H}_{2}$ such that
$[\xi ,\eta ]=\u27e8T\xi ,\eta \u27e9,\xi \in {H}_{1},\eta \in {H}_{2}.$ 
Thus, there is a correspondence between bounded linear operators and bounded sesquilinear forms. Actually, in the early twentieth century, spectral theory was formulated solely in terms of sesquilinear forms on Hilbert spaces. Only later it was realized that this could be achieved, perhaps in a more intuitive manner, by considering linear operators^{} instead. The linear operator approach has its advantages, as for example one can define the composition of linear operators but not of sesquilinear forms. Nevertheless it is many times useful to define a linear operator by specifying its sesquilinear form.
Title  Riesz representation theorem of bounded sesquilinear forms 

Canonical name  RieszRepresentationTheoremOfBoundedSesquilinearForms 
Date of creation  20130322 18:41:38 
Last modified on  20130322 18:41:38 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  10 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 47A07 
Classification  msc 46C05 
Synonym  Riesz lemma on bounded sesquilinear forms 
Synonym  correspondence between bounded operators and bounded sesquilinear forms 
Defines  bounded sesquilinear form 