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Don't let the bad wording of the paper fool you. The statement of lemma 2 in [1] above does not means that the functional $G$ constructed by the kernel $g_0$ separates all the points $u, v\in K_-$ such that $u\neq v$: it means only that for any such two functions you can construct a kernel such that the associated linear functional constructed as above is such that $Gu\neq Gv$. At first I was fooled by the same thought but then I read Remark 4 in the same page ([1], §IV p. 1153), which contains the statement quoted below

Suppose $E$ is a compact metric space and $\mathbf G$ a set of continuous functionals on $E$ which separate points, that is, for any distinct $u,v\in E$ there is a $G\in\mathbf G$ such that $Gu \neq Gv$.

Then I simply realized that the authors want only to show that the functional $$\DeclareMathOperator{\dmu}{d\!}G_0(q)=\int\limits_0^{+\infty}g_0(\tau)q(-\tau)\dmu\tau \quad \forall q\in K_- $$ with the kernel $g_0$ constructed as described above is such that $G_0 u\neq G_0v$ for any fixed couple of different functions chosen for its construction. If $u_1, v_1\in K_-$ are a different couple such that $u_1\neq v_1$, you do not know if $G_0u_1\neq G_0 v_1$ (an in general this is not true): however you can construct another functional, say $G_1$, whose kernel is defined as exactly as $g_0$ i.e. $$ g_1(t):= [u_1(-t)-v_1(-t)] w(t)\exp (-t) $$ and separates the new points but obviously possibly not the points $u$ and $v$.

Don't let the bad wording of the paper fool you. The statement of lemma 2 in [1] above does not mean that the functional $G$ constructed by the kernel $g_0$ separates all the points $u, v\in K_-$ such that $u\neq v$: it means only that, for any such two functions, you can construct a kernel such that the associated linear functional constructed as above is such that $Gu\neq Gv$. At first I was fooled by the same thought but then I read Remark 4 in the same page ([1], §IV p. 1153), which contains the statement quoted below

Suppose $E$ is a compact metric space and $\mathbf G$ a set of continuous functionals on $E$ which separate points, that is, for any distinct $u,v\in E$ there is a $G\in\mathbf G$ such that $Gu \neq Gv$.

Then I simply realized that the authors want only to show that the functional $$\DeclareMathOperator{\dmu}{d\!}G_0(q)=\int\limits_0^{+\infty}g_0(\tau)q(-\tau)\dmu\tau \quad \forall q\in K_- $$ with the kernel $g_0$ constructed as described above is such that $G_0 u\neq G_0v$ for any fixed couple of different functions chosen for its construction. If $u_1, v_1\in K_-$ are a different couple such that $u_1\neq v_1$, you do not know if $G_0u_1\neq G_0 v_1$ (an in general this is not true): however you can construct another functional, say $G_1$, whose kernel is defined as exactly as $g_0$ i.e. $$ g_1(t):= [u_1(-t)-v_1(-t)] w(t)\exp (-t) $$ and separates the new points but obviously possibly not the points $u$ and $v$.

Don't let the bad wording of the paper fool you. The statement of lemma 2 in [1] above does not signify that the functional $G$ constructed by the kernel $g_0$ separates all the points $u, v\in K_-$ such that $u\neq v$: it means only that for any such two functions you can construct a kernel such that the associated linear functional constructed as above is such that $Gu\neq Gv$. At first I was fooled by the same thought but then I read Remark 4 in the same page ([1], §IV p. 1153), which contains the statement quoted below

Suppose $E$ is a compact metric space and $G$ a set of continuous functionals on $E$ which separate points, that is, for any distinct $u,v\in E$ there is a $G\in\mathbf G$ such that $Gu \neq Gv$.

Then I simply realized that the authors want only to show that the functional $$\DeclareMathOperator{\dmu}{d\!}G_0(q)=\int\limits_0^{+\infty}g_0(\tau)q(-\tau)\dmu\tau \quad \forall q\in K_- $$ with the kernel $g_0$ constructed as described above is such that $G_0 u\neq G_0v$ for any fixed couple of different functions chosen for its construction. If $u_1, v_1\in K_-$ are a different couple such that $u_1\neq v_1$, you do not know if $G_0u_1\neq G_0 v_1$ (an in general this is not true): however you can construct another functional, say $G_1$, whose kernel is defined as exactly as $g_0$ i.e. $$ g_1(t):= [u_1(-t)-v_1(-t)] w(t)\exp (-t) $$ and separates the new points but obviously possibly not the points $u$ and $v$.

Don't let the bad wording of the paper fool you. The statement of lemma 2 in [1] above does not means that the functional $G$ constructed by the kernel $g_0$ separates all the points $u, v\in K_-$ such that $u\neq v$: it means only that for any such two functions you can construct a kernel such that the associated linear functional constructed as above is such that $Gu\neq Gv$. At first I was fooled by the same thought but then I read Remark 4 in the same page ([1], §IV p. 1153), which contains the statement quoted below

Suppose $E$ is a compact metric space and $\mathbf G$ a set of continuous functionals on $E$ which separate points, that is, for any distinct $u,v\in E$ there is a $G\in\mathbf G$ such that $Gu \neq Gv$.

Then I simply realized that the authors want only to show that the functional $$\DeclareMathOperator{\dmu}{d\!}G_0(q)=\int\limits_0^{+\infty}g_0(\tau)q(-\tau)\dmu\tau \quad \forall q\in K_- $$ with the kernel $g_0$ constructed as described above is such that $G_0 u\neq G_0v$ for any fixed couple of different functions chosen for its construction. If $u_1, v_1\in K_-$ are a different couple such that $u_1\neq v_1$, you do not know if $G_0u_1\neq G_0 v_1$ (an in general this is not true): however you can construct another functional, say $G_1$, whose kernel is defined as exactly as $g_0$ i.e. $$ g_1(t):= [u_1(-t)-v_1(-t)] w(t)\exp (-t) $$ and separates the new points but obviously possibly not the points $u$ and $v$.

Don't let the bad wording of the paper fool you. The statement of lemma 2 in [1] above does not signify that the functional $G$ constructed by the kernel $g_0$ separates all the points $u, v\in K_-$ such that $u\neq v$: it means only that for any such two functions you can construct a kernel such that the associated linear functional constructed as above is such that $Gu\neq Gv$. At first I was fooled by the same thought but then I read Remark 4 in the same page ([1], §IV p. 1153), which cotntains the statement quoted below

Suppose $E$ is a compact metric space and $G$ a set of continuous functionals on $E$ which separate points, that is, for any distinct $u,v\in E$ there is a $G\in\mathbf G$ such that $Gu \neq Gv$.

Then I simply realized that the authors want only to show that the functional $$\DeclareMathOperator{\dmu}{d\!}G_0(q)=\int\limits_0^{+\infty}g_0(\tau)q(-\tau)\dmu\tau \quad \forall q\in K_- $$ with the kernel $g_0$ constructed as described above is such that $G_0 u\neq G_0v$ for any fixed couple of different functions chosen for its construction. If $u_1, v_1\in K_-$ are a different couple such that $u_1\neq v_1$, you do not know if $G_0u_1\neq G_0 v_1$ (an in general this is not true): however you can construct another functional, say $G_1$, whose kernel is defined as exactly as $g_0$ i.e. $$ g_1(t):= [u_1(-t)-v_1(-t)] w(t)\exp (-t) $$ and separates the new points but obviously possibly not the points $u$ and $v$.

Don't let the bad wording of the paper fool you. The statement of lemma 2 in [1] above does not signify that the functional $G$ constructed by the kernel $g_0$ separates all the points $u, v\in K_-$ such that $u\neq v$: it means only that for any such two functions you can construct a kernel such that the associated linear functional constructed as above is such that $Gu\neq Gv$. At first I was fooled by the same thought but then I read Remark 4 in the same page ([1], §IV p. 1153), which contains the statement quoted below

Suppose $E$ is a compact metric space and $G$ a set of continuous functionals on $E$ which separate points, that is, for any distinct $u,v\in E$ there is a $G\in\mathbf G$ such that $Gu \neq Gv$.

Then I simply realized that the authors want only to show that the functional $$\DeclareMathOperator{\dmu}{d\!}G_0(q)=\int\limits_0^{+\infty}g_0(\tau)q(-\tau)\dmu\tau \quad \forall q\in K_- $$ with the kernel $g_0$ constructed as described above is such that $G_0 u\neq G_0v$ for any fixed couple of different functions chosen for its construction. If $u_1, v_1\in K_-$ are a different couple such that $u_1\neq v_1$, you do not know if $G_0u_1\neq G_0 v_1$ (an in general this is not true): however you can construct another functional, say $G_1$, whose kernel is defined as exactly as $g_0$ i.e. $$ g_1(t):= [u_1(-t)-v_1(-t)] w(t)\exp (-t) $$ and separates the new points but obviously possibly not the points $u$ and $v$.