Skip to main content Contents Prev Up Next \(\DeclareMathOperator{\RE}{Re}
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Section 2.4 Exercises
Exercises Exercises
1. Prove that
(2.3.1) is an an inner product on
\(RL^2\text{.}\) (Hint: parametrize
\(\T\) by
\(z = e^{i\theta}\) for
\(-\pi \lt \theta \leq \pi\) ).
2. Prove that
(2.3.2) is an inner product on
\(W[a,b]\text{.}\)
3. Prove that
(2.3.3) is an inner product on
\(TP\text{.}\)
4.
Prove the Pythagorean theorem in an inner product space \(V\text{;}\) that is, show that if \(\ip{x}{y} = 0\) for \(x, y \in V\) then
\begin{equation*}
\norm{x}^2 + \norm{y}^2 = \norm{x + y}^2.
\end{equation*}
5.
Define the elementary functions
\begin{equation*}
e_n(t) = e^{2\pi i n t}
\end{equation*}
where \(n \in \Z\) and for values of \(t \in [0,1]\text{.}\)
Prove that
\(e_n \perp e_m\) when
\(m \neq n\) as elements of
\(C[0,1]\) and as elements of
\(W[0,1]\) (that is, with respect to
(2.1.1) and
(2.3.2) ). (This is precursor work to the development of
orthonormal systems , which we will meet shortly.)
