Consider space Hilbert space L^2[0 1] of square integrable functions on say [0 1]. Saying more carefully, there is one subtle detail: this space is defined as factor space by the functions which are have norm zero, in particular if we change value of f(x) at one point, then in this factor space the two functions will be still the same. So formally speaking the value at points of elements from $L^2[0 1]$ is not defined.

Take point x in [0 1].

Continuous function can be evaluated at x: ev_x (f) = f(x) .

Question Can this linear functional be extended from continuous functions to the L^2[0 1] ? I.e. Does there exists a continuous linear functional $\Psi:L^2[0 1]\to R$, such that it coincide with $ev_x$ on the subspace of continuous functions.

Reformulation: $i: C[0 1] \to L^2[0 1]$ consider dual map: $i^*: (L^2)^* [0 1] \to C^*[0 1] $ ,

dual spaces are known - for $C[0 1]$ it is space of functions of bounded variation, $(L^2)^*[0 1] = L^2[0 1] $. So we can ask does $ev_x$ lies in the image of this map.

Motivation: in physics "coherent states" defined by this idea, (as far as I understand), e.g. approach by Rawnsley. But they are defined using "holomorphic polarization", where of course we can take value of holomorphic function at a point. So I am puzzled wether we really need holomorphic polarization.

Consider the Hilbert space $L^2[0, 1]$ of square integrable functions on $[0, 1]$. Saying more carefully, there is one subtle detail: this space is defined as factor space by the functions which are have norm zero, in particular if we change value of $f(x)$ at one point, then in this factor space the two functions will be still the same. So formally speaking the value at points of elements from $L^2[0 1]$ is not defined.

Take point $x$ in $[0, 1]$. A continuous function $f$ can be evaluated at $x$: $ev_x (f) = f(x)$.

Question: Can the linear functional $ev_x$ be extended from continuous functions to $L^2[0, 1]$? In other words, does there exists a continuous linear functional $\Psi:L^2[0, 1]\to \mathbb{R}$, such that it coincides with $ev_x$ on the subspace of continuous functions?

Another way to Reformulation: let $i: C[0, 1] \to L^2[0, 1]$ be the inclusion and consider the dual map $i : (L^2[0,1])^{ * } \to (C[0,1])^{ * }$. The dual spaces involved are known: for $C[0, 1]$ it is space of functions of bounded variation, whereas $(L^2[0, 1])^{*} = L^2[0, 1]$. So we can ask whether $ev_x$ lies in the image of $i^{ * }$.

Motivation: in physics "coherent states" defined by this idea, (as far as I understand), e.g. approach by Rawnsley. But they are defined using "holomorphic polarization", where of course we can take value of holomorphic function at a point. So I am puzzled wether we really need holomorphic polarization.

Consider space Hilbert space L^2[0 1] of square integrable functions on say [0 1]. Saying more carefully, there is one subtle detail: this space is defined as factor space by the functions which are have norm zero, in particular if we change value of f(x) at one point, then in this factor space the two functions will be still the same. So formally speaking the value at points of elements from $L^2[0 1]$ is not defined.

Take point x in [0 1].

Continuous function can be evaluated at x: ev_x (f) = f(x) .

Question Can this linear functional be extended from continuous functions to the L^2[0 1] ? I.e. Does there exists a continuous linear functional $\Psi:L^2[0 1]-> R$, such that it coincide with $ev_x$ on the subspace of continuous functions.

Reformulation: $i: C[0 1]-> L^2[0 1]$ consider dual map: $i^*: (L^2)^* [0 1] -> C^*[0 1] $ ,

dual spaces are known - for $C[0 1]$ it is space of functions of bounded variation, $(L^2)^*[0 1] = L^2[0 1] $. So we can ask does $ev_x$ lies in the image of this map.

Motivation: in physics "coherent states" defined by this idea, (as far as I understand), e.g. approach by Rawnsley. But they are defined using "holomorphic polarization", where of course we can take value of holomorphic function at a point. So I am puzzled wether we really need holomorphic polarization.

Consider space Hilbert space L^2[0 1] of square integrable functions on say [0 1]. Saying more carefully, there is one subtle detail: this space is defined as factor space by the functions which are have norm zero, in particular if we change value of f(x) at one point, then in this factor space the two functions will be still the same. So formally speaking the value at points of elements from $L^2[0 1]$ is not defined.

Take point x in [0 1].

Continuous function can be evaluated at x: ev_x (f) = f(x) .

Question Can this linear functional be extended from continuous functions to the L^2[0 1] ? I.e. Does there exists a continuous linear functional $\Psi:L^2[0 1]\to R$, such that it coincide with $ev_x$ on the subspace of continuous functions.

Reformulation: $i: C[0 1] \to L^2[0 1]$ consider dual map: $i^*: (L^2)^* [0 1] \to C^*[0 1] $ ,

dual spaces are known - for $C[0 1]$ it is space of functions of bounded variation, $(L^2)^*[0 1] = L^2[0 1] $. So we can ask does $ev_x$ lies in the image of this map.

Motivation: in physics "coherent states" defined by this idea, (as far as I understand), e.g. approach by Rawnsley. But they are defined using "holomorphic polarization", where of course we can take value of holomorphic function at a point. So I am puzzled wether we really need holomorphic polarization.