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cardfrontfoot{Functional Analysis}


begin{flashcard}[Definition]{Norm on a Linear Space \ Normed Space}

A real-valued function $||x||$ defined on a linear space $X$, where
$x in X$, is said to be a emph{norm on} $X$ if


smallskip

begin{description}
item [Positivity] $||x|| geq 0$,
item [Triangle Inequality] $||x+y|| leq ||x|| + ||y||$,
item [Homogeneity] $||alpha x|| = |alpha| : ||x||$,
$alpha$ an arbitrary scalar,
item [Positive Definiteness] $||x|| = 0$ if and only if $x=0$,
end{description}

smallskip

where $x$ and $y$ are arbitrary points in $X$.

medskip

A linear/vector space with a norm is called a emph{normed space}.
end{flashcard}

begin{flashcard}[Definition]{Inner Product}

Let $X$ be a complex linear space. An emph{inner product} on $X$ is
a mapping that associates to each pair of vectors $x$, $y$ a scalar,
denoted $(x,y)$, that satisfies the following properties:

medskip

begin{description}
item [Additivity] $(x+y,z) = (x,z) + (y,z)$,
item [Homogeneity] $(alpha : x, y) = alpha (x,y)$,
item [Symmetry] $(x,y) = overline{(y,x)}$,
item [Positive Definiteness] $(x,x) > 0$, when $xneq0$.
end{description}
end{flashcard}


begin{flashcard}[Definition]{Linear Transformation/Operator}

A transformation $L$ of (operator on) a linear space $X$ into a linear
space $Y$, where $X$ and $Y$ have the same scalar field, is said to be
a emph{linear transformation (operator)} if

medskip

begin{enumerate}
item $L(alpha x) = alpha L(x), forall xin X$ and $forall$
scalars $alpha$, and
item $L(x_1 + x_2) = L(x_1) + L(x_2)$ for all $x_1,x_2 in X$.
end{enumerate}

end{flashcard}
end{document}

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