Let \(V\) be the real inner product space consisting of the space of real-valued continuous functions on the interval, \(-1 \leq t \leq 1\), with the inner product \[ (f \mid g) = \int_{-1}^{1} f(t)g(t)\,dt. \] Let \(W\) be the subspace of odd functions, i.e., functions satisfying \(f(-t) = -f(t)\). Find the orthogonal complement of \(W\).
Solution
1. Define the subspace of even functions \(W_{\text{even}} = \{g \in V : g(-t) = g(t) \text{ for all } t \in [-1, 1]\}\).
definition.
2. Show that \(W_{\text{even}} \subseteq W^\perp\):
2.1. Let \(g \in W_{\text{even}}\) and \(f \in W\).
setup.
2.2. The integrand \(h(t) = f(t)g(t)\) is an odd function:\[ h(-t) = f(-t)g(-t) = (-f(t))g(t) = -f(t)g(t) = -h(t). \]
\(f\) is odd and \(g\) is even.
2.3. The integral of any continuous odd function over a symmetric interval \([-1, 1]\) is zero:\[ (f \mid g) = \int_{-1}^1 f(t)g(t)\,dt = \int_{-1}^1 h(t)\,dt = 0. \]
substitution \(u = -t\) gives \(\int_{-1}^1 h(t)\,dt = -\int_{-1}^1 h(u)\,du\).
2.4. Hence \(g \in W^\perp\), so \(W_{\text{even}} \subseteq W^\perp\).
3 holds for all \(f \in W\).
3. Show that \(W^\perp \subseteq W_{\text{even}}\):
3.1. Let \(g \in W^\perp\), so \((f \mid g) = 0\) for all \(f \in W\).
definition of orthogonal complement.
3.2. Decompose \(g\) into its even and odd parts:\[ g(t) = g_{\text{even}}(t) + g_{\text{odd}}(t), \quad \text{where } g_{\text{odd}}(t) = \frac{g(t) - g(-t)}{2} \text{ and } g_{\text{even}}(t) = \frac{g(t) + g(-t)}{2}. \]
algebraic identity and standard parity decomposition of functions.
3.3. \(g_{\text{odd}} \in W\) and \(g_{\text{even}} \in W_{\text{even}}\).
\(g_{\text{odd}}(-t) = -g_{\text{odd}}(t)\) and \(g_{\text{even}}(-t) = g_{\text{even}}(t)\), with continuity preserved under linear combinations.
3.4. Since \(g \in W^\perp\) and \(g_{\text{odd}} \in W\), \((g_{\text{odd}} \mid g) = 0\).
3.1.
3.5. By 2, \(g_{\text{even}} \in W_{\text{even}} \subseteq W^\perp\), so \((g_{\text{odd}} \mid g_{\text{even}}) = 0\).
2.
2.6. Expand the inner product:
\[ 0 = (g_{\text{odd}} \mid g) = (g_{\text{odd}} \mid g_{\text{even}} + g_{\text{odd}}) = (g_{\text{odd}} \mid g_{\text{even}}) + (g_{\text{odd}} \mid g_{\text{odd}}) = \|g_{\text{odd}}\|^2. \]
2.5.
2.7. \(\|g_{\text{odd}}\|^2 = \int_{-1}^1 (g_{\text{odd}}(t))^2\,dt = 0\).
definition of the inner product norm.
2.8. Since \(g_{\text{odd}}\) is continuous and \((g_{\text{odd}}(t))^2 \ge 0\), \((g_{\text{odd}}(t))^2 = 0\) for all \(t \in [-1, 1]\), so \(g_{\text{odd}}(t) = 0\) identically.
a continuous nonnegative function with zero integral is identically zero.
2.9. Thus \(g(t) = g_{\text{even}}(t) \in W_{\text{even}}\), so \(W^\perp \subseteq W_{\text{even}}\).
2 and 8.
4. Conclusion: \(W^\perp = W_{\text{even}}\) is the subspace of all real-valued continuous even functions on \([-1, 1]\).
2 and 3.
Q.E.D.