$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Self-defined math definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Math symbol commands \newcommand{\intd}{\,{\rm d}} % Symbol 'd' used in integration, such as 'dx' \newcommand{\diff}{{\rm d}} % Symbol 'd' used in differentiation \newcommand{\Diff}{{\rm D}} % Symbol 'D' used in differentiation \newcommand{\pdiff}{\partial} % Partial derivative \newcommand{DD}[2]{\frac{\diff}{\diff #2}\left( #1 \right)} \newcommand{Dd}[2]{\frac{\diff #1}{\diff #2}} \newcommand{PD}[2]{\frac{\pdiff}{\pdiff #2}\left( #1 \right)} \newcommand{Pd}[2]{\frac{\pdiff #1}{\pdiff #2}} \newcommand{\rme}{{\rm e}} % Exponential e \newcommand{\rmi}{{\rm i}} % Imaginary unit i \newcommand{\rmj}{{\rm j}} % Imaginary unit j \newcommand{\vect}[1]{\boldsymbol{#1}} % Vector typeset in bold and italic \newcommand{\phs}[1]{\dot{#1}} % Scalar phasor \newcommand{\phsvect}[1]{\boldsymbol{\dot{#1}}} % Vector phasor \newcommand{\normvect}{\vect{n}} % Normal vector: n \newcommand{\dform}[1]{\overset{\rightharpoonup}{\boldsymbol{#1}}} % Vector for differential form \newcommand{\cochain}[1]{\overset{\rightharpoonup}{#1}} % Vector for cochain \newcommand{\bigabs}[1]{\bigg\lvert#1\bigg\rvert} % Absolute value (single big vertical bar) \newcommand{\Abs}[1]{\big\lvert#1\big\rvert} % Absolute value (single big vertical bar) \newcommand{\abs}[1]{\lvert#1\rvert} % Absolute value (single vertical bar) \newcommand{\bignorm}[1]{\bigg\lVert#1\bigg\rVert} % Norm (double big vertical bar) \newcommand{\Norm}[1]{\big\lVert#1\big\rVert} % Norm (double big vertical bar) \newcommand{\norm}[1]{\lVert#1\rVert} % Norm (double vertical bar) \newcommand{\ouset}[3]{\overset{#3}{\underset{#2}{#1}}} % over and under set % Super/subscript for column index of a matrix, which is used in tensor analysis. \newcommand{\cscript}[1]{\;\; #1} % Star symbol used as prefix in front of a paragraph with no indent \newcommand{\prefstar}{\noindent$\ast$ } % Big vertical line restricting the function. % Example: $u(x)\restrict_{\Omega_0}$ \newcommand{\restrict}{\big\vert} % Math operators which are typeset in Roman font \DeclareMathOperator{\sgn}{sgn} % Sign function \DeclareMathOperator{\erf}{erf} % Error function \DeclareMathOperator{\Bd}{Bd} % Boundary of a set, used in topology \DeclareMathOperator{\Int}{Int} % Interior of a set, used in topology \DeclareMathOperator{\rank}{rank} % Rank of a matrix \DeclareMathOperator{\divergence}{div} % Curl \DeclareMathOperator{\curl}{curl} % Curl \DeclareMathOperator{\grad}{grad} % Gradient \DeclareMathOperator{\tr}{tr} % Trace \DeclareMathOperator{\span}{span} % Span $$

止于至善

As regards numerical analysis and mathematical electromagnetism

Kernels and image sets for an operator and its dual

Let $\mathcal{X}$ and $\mathcal{Y}$ be Hilbert spaces. Let $A: \mathcal{X} \rightarrow \mathcal{Y}$ be a bounded and linear operator. Then

$$
A(\mathcal{X})^{\perp} = \ker(A^*), \; \ker(A^*)^{\perp}=\overline{A(\mathcal{X})}.
$$

This relation can be illustrated in the following figure. Here we use blue color to represent the "effective mapping", i.e. from the orthogonal complement of the operator's  kernel to the operator's image set. And red color is used to represent the mapping from the operator's kernel to the zero element of the operator's range space. The orthogonal complement of a subset in a Hilbert space is a subspace, which is closed. Meanwhile, the image set of an operator is not necessarily closed. This is why there is a closure operation in $\ker(A^*)^{\perp}=\overline{A(\mathcal{X})}$.

posted @ 2019-05-18 23:10  皮波迪博士  阅读(209)  评论(0编辑  收藏  举报