% Lecture notes for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2011/11/3

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\chapter{Hilbert spaces}

\section{Hilbert spaces}

\subsection{Vector spaces}

A \keyword{vector space} is a set whose elements, called \keyword{vectors}, can be added, and multiplied by a scalar, in the usual way.
We will focus on complex vector spaces, in which case the scalars are complex numbers.

We will denote a vector $\phi$ by $\ket\phi$, and a scalar $\alpha$ by $\alpha$. Then we have the basic operations
\begin{equation}
\ket\phi + \ket\psi = \ket\chi
\end{equation}
and
\begin{equation}
\alpha \ket\phi = \ket\xi
\end{equation}

\subsection{Hermitian conjugation and covectors}

It is natural to extend the concept of complex conjugation to vectors, in which case it is called \keyword{Hermitian conjugation} and denoted\footnote{Mathematicians simply use a superscript $*$.} by a superscript $\dagger$.
The Hermitian conjugate of a vector $\ket\phi$ is a \keyword{covector} $\bra\phi$, and vice versa
\begin{equation}
\ket\phi^\dagger = \bra\phi
\quad , \quad
\bra\phi^\dagger = \ket\phi
\end{equation}
The covectors form a dual vector space, with Hermitian conjugation providing an antilinear bijection between the vectors and the covectors
\begin{equation}
\left( \alpha \ket\phi + \beta \ket\psi \right)^\dagger = \alpha^* \bra\phi + \beta^* \bra\psi
\end{equation}

Vectors and covectors can be \keyword{contracted} together to give a scalar
\begin{equation}
\braket\phi\psi \in \mathbb{C}
\end{equation}
with
\begin{equation}
\bra\chi \left( \alpha \ket\phi + \beta \ket\psi \right) = \alpha \braket\chi\phi + \beta \braket\chi\psi
\end{equation}
and
\begin{equation}
\braket\psi\phi
= \left( \braket\phi\psi  \right)^\dagger
= \left( \braket\phi\psi \right)^*
\end{equation}

Combining Hermitian conjugation with contraction gives the \keyword{inner product} of two vectors
\begin{equation}
\left( \ket\phi , \ket\psi \right)
= \ket\phi^\dagger \ket\psi
= \braket\phi\psi
\end{equation}
The magnitude squared of a vector is
\begin{equation}
|\phi|^2 = \braket\phi\phi
\end{equation}
An inner product on a complex vector space has $|\phi|^2 > 0$ unless $\ket\phi = 0$.

A \keyword{Hilbert space} is a vector space with an inner product, which is complete, i.e.\ which contains the limit of any convergent sequence of vectors.
Completeness is important because it allows us to use calculus.

\subsection{Subspaces}

A \keyword{subspace} $\hilbert{G}$ of a Hilbert space $\hilbert{H}$, denoted $\hilbert{G} \subset \hilbert{H}$, is a subset of $\hilbert{H}$ which is also a Hilbert space.
The intersection of subspaces, $\hilbert{F} \cap \hilbert{G}$, is also a subspace, but the union, $\hilbert{F} \cup \hilbert{G}$, is not.
Instead subspaces can be added
\begin{equation}
\hilbert{F} + \hilbert{G} = \left\{ \ket{f} + \ket{g} : \ket{f} \in \hilbert{F} , \ket{g} \in \hilbert{G} \right\}
\end{equation}
If $\hilbert{H} = \hilbert{F} + \hilbert{G}$ and $\hilbert{F} \cap \hilbert{G} = \{0\}$ then $\hilbert{H}$ is the \keyword{direct sum} of $\hilbert{F}$ and $\hilbert{G}$, $\hilbert{H} = \hilbert{F} \oplus \hilbert{G}$, and $\hilbert{F}$ and $\hilbert{G}$ are complementary within $\hilbert{H}$.
A complement is not unique but we can use the inner product to define a unique \keyword{orthogonal complement} $\hilbert{G}^\perp$ of a subpsace $\hilbert{G} \subset \hilbert{H}$
\begin{equation}
\hilbert{G}^\perp = \left\{ \ket{h} : \braket{h}{g} = 0 \textrm{ for all } \ket{g} \in \hilbert{G} \right\}
\end{equation}

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