Question

Let $L^2(0, 1)$ be the space of integrable functions $f : (0, 1) ightarrow mathbb{R}$ such that egin{equation} int_0^1 f(t)^2 dt < infty. end{equation} 1) Show that egin{equation} langle f|g angle = int_0^1 f(t) g(t) dt end{equation} defines an inner product on $L^2(0, 1)$ ewline 2 show that $mathcal{B} = {sin(2pi nt)}_{n in mathbb{N}}$ is a family ewline of orthogonal vectors with respect to this scalar product.

          Let $L^2(0, 1)$ be the space of integrable functions $f : (0, 1) 
ightarrow mathbb{R}$ such that egin{equation*} int_0^1 f(t)^2 dt < infty. end{equation*} 1) Show that egin{equation*} langle f|g 
angle = int_0^1 f(t) g(t) dt end{equation*} defines an inner product on $L^2(0, 1)$ 
ewline 2 show that $mathcal{B} = {sin(2pi nt)}_{n in mathbb{N}}$ is a family 
ewline of orthogonal vectors with respect to this scalar product.

Let L^2(0, 1) be the space of integrable functions f : (0, 1)
ightarrow mathbbR such that eginequation* int0^1 f(t)^2 dt < infty. endequation* 1) Show that eginequation* langle f|g
angle = int0^1 f(t) g(t) dt endequation* defines an inner product on L^2(0, 1)
ewline 2 show that mathcalB = sin(2pi nt)n in mathbbN is a family
ewline of orthogonal vectors with respect to this scalar product.

Added by John S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Thanks for your feedback!

Let L2(0, 1) be the space of integrable functions f : (0, 1) → R such that ∫01 |f(t)|² dt < ∞. Show that ⟨f,g⟩ = ∫01 f(t)g(t) dt defines an inner product on L²(0, 1). Show that B = {sin(2πnt)}n∈N is a family of orthogonal vectors with respect to this scalar product.