grandes-ecoles 2010 QI.A.1

grandes-ecoles · France · centrale-maths2__mp Sequences and Series Inner Product Spaces, Orthogonality, and Hilbert Space Structure on Sequence/Function Spaces

For every element $x$ of $E$, we denote by $h(x)$ the application from $E$ to $E$ such that $\forall y \in E, h(x)(y) = \varphi(x,y)$.
a) Show that, for all $x$ in $E$, $h(x)$ is an element of the dual of $E$, denoted $E^{*}$.
b) Show that $h$ is a linear application from $E$ to $E^{*}$.

For every element $x$ of $E$, we denote by $h(x)$ the application from $E$ to $E$ such that $\forall y \in E, h(x)(y) = \varphi(x,y)$.

a) Show that, for all $x$ in $E$, $h(x)$ is an element of the dual of $E$, denoted $E^{*}$.

b) Show that $h$ is a linear application from $E$ to $E^{*}$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.B.3 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4