Question

Example 8.8 Let $F[a, b] = \{f(x)\}$ be the set of all functions $f: [a, b] \to \mathbb{R}$, where $[a, b] = \{x \in \mathbb{R} : a \le x \le b\}$. Show that $\langle f, g \rangle = \int_a^b f(x)g(x) dx$ is an inner product on $F[a, b]$.

          Example 8.8 Let $F[a, b] = \{f(x)\}$ be the set of all functions
$f: [a, b] \to \mathbb{R}$,
where $[a, b] = \{x \in \mathbb{R} : a \le x \le b\}$. Show that
$\langle f, g \rangle = \int_a^b f(x)g(x) dx$
is an inner product on $F[a, b]$.

$Example 8.8 Let F[a, b] = {f(x)} be the set of all functions f: [a, b] →ℝ, where [a, b] = {x ∈ℝ : a ≤ x ≤ b}. Show that ⟨ f, g ⟩ = ^b f(x)g(x) dx is an inner product on F[a, b].$

Added by Anthony B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/16/2023

Step 1

To show that the inner product ⟨f, g⟩ = ∫_a^b f(x)g(x)dx is symmetric, we need to show that ⟨f, g⟩ = ⟨g, f⟩ for all functions f, g in F[a, b]. ∫_a^b f(x)g(x)dx = ∫_a^b g(x)f(x)dx (by commutativity of multiplication) Therefore, the inner product is symmetric. Show more…

Show all steps

Thanks for your feedback!

Example 8.8: Let F[a, b] = {f(x)} be the set of all functions f: [a, b] → R, where [a, b] = {x in R: a ≤ x ≤ b}. Show that ⟨f, g⟩ = ∫_a^b f(x)g(x)dx is an inner product on F[a, b]. Let f: [a, b] → R, where [a, b] = {x in R: a < x < b}. Show that ⟨f, g⟩ = ∫_a^b f(x)g(x)dx is an inner product on F[a, b].