Question

a. Let $\mathbf{w}\left(t\right)$be a positive-valued function in $\mathbf{C}\mathbf{[}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{]}$, where $\mathit{b}\mathbf{>}\mathit{a}$. Verify that the rule $\left(f,g\right)\mathbf{=}{\mathbf{\int }}_{\mathbf{a}}^{\mathbf{b}}\mathbf{w}\left(t\right)\mathbf{f}\left(t\right)\mathbf{g}\left(t\right)\mathbf{dt}$defines an inner product on $\mathbf{C}\left[a,b\right]$.

b. If we chose the weight function $\mathbf{w}\left(t\right)$so that ${\mathbf{\int }}_{\mathbf{a}}^{\mathbf{b}}\mathbf{w}\left(t\right)\mathbf{dt}\mathbf{=}\mathbf{1}$, what is the norm of the constant function $\mathbf{f}\left(t\right)\mathbf{=}\mathbf{1}$in this inner product space?

Short answer

(a) The value of $\left(f,g\right)$defines an inner product on $C\left[a,b\right]$is $\left(g,f\right)$.

(b) The value of the norm of the constant function $f\left(t\right)=1$defines an inner product on $C\left[a,b\right]$.

Step-by-step solution

Write the given data from the question.

Let $\mathrm{w}\left(\mathrm{t}\right)$be a positive-valued function in $\mathrm{C}\left[\mathrm{a},\mathrm{b}\right]$, where b>a.

Determine the formula of defines an inner product on C[a,b] and the norm of the constant function f(t)=1 defines an inner product on C[a,b].

Write the formula of defines an inner product on$C\left[a,b\right]$.

$\left(\mathrm{f},\mathrm{g}\right)={\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{w}\left(\mathrm{t}\right)\mathrm{f}\left(\mathrm{t}\right)\mathrm{d}\left(\mathrm{t}\right)\mathrm{dt}$ …… (1)

Here,$\mathrm{w}\left(\mathrm{t}\right)$is positive-valued function in $C\left[a,b\right]$,$f\left(t\right)$is constant function and $g\left(t\right)$is defines an inner product.

Write the formula of the norm of the constant function $f\left(t\right)=1$defines an inner product on $C\left[a,b\right]$.

$\mathrm{f}\left(\mathrm{t}\right)=\left(\mathrm{f},\mathrm{f}\right)$

Here, $\left(\mathrm{f},\mathrm{f}\right)$is defines at inner function.

(a) Determine the value of the norm of the constant function f(t)=1 defines an inner product on C[a, b].

This is done by direct verification. Let $f,g,h\in C[a,b]$and $c\in R$.

(1)

Determine the value of $(f,g)$defines an inner product on $C\left[a,b\right]$.

Substitute $w\left(t\right)$for $w\left(t\right)$, $f\left(t\right)$for $g\left(t\right)$and $g\left(t\right)$for $f\left(t\right)$into equation (1).

$$\begin{gathered} \{f,g\}={\int }_a^{b}W\left(t\right)g\left(t\right)f\left(t\right)dt \\ =\{g,f\} \end{gathered}$$

Determine the value of$\left(f+h,h\right)$.

$$\begin{gathered} \left(f+h,g\right)={\int }_a^{b}w\left(t\right)f\left(t\right)+h\left(t\right)g\left(t\right)dt \\ ={\int }_a^{b}w\left(t\right)f\left(t\right)+h\left(t\right)g\left(t\right)+{\int }_a^{b}w\left(t\right)h\left(t\right)g\left(t\right)dt \\ =\left(f,g\right)+\left(h,g\right) \end{gathered}$$

Determine the$\left(cf,g\right)$.

$$\begin{gathered} \left(cf,g\right)={\int }_a^{b}w\left(t\right)cf\left(t\right)g\left(t\right)dt \\ =c{\int }_a^{b}w\left(t\right)f\left(t\right)g\left(t\right)dt \\ =c\left(f,g\right) \end{gathered}$$

Determine the$\left(f,f\right)$.

$$\begin{gathered} \left(f,f\right)={\int }_a^{b}w\left(t\right)f\left(t\right)f\left(t\right)dt \\ ={\int }_a^{b}w\left(t\right)f{\left(t\right)}^{2}dt>0 \end{gathered}$$

Because $w\left(t\right)=0$on $\left[a,b\right]$ and f is a nonzero continuous function, so there must be some interval in $\left[a,b\right]$on which $f^2>0$.

The axioms $\left(1\right)$, $\left(2\right)$, $\left(3\right)$, $\left(4\right)$hold for all $f,g,h\in C[a,b]$,$c\in \mathrm{ℝ}$.

Therefore, the value of $\left(f,g\right)$defines an inner product on $C\left[a,b\right]$is $\left(g,f\right)$.

(b) Determine the value of the norm of the constant function f(t)=1  defines an inner product on C[a,b].

Assume $f\left(t\right)=1$, ${\int }_a^{b}w\left(t\right)dt=1$.

Determine the norm of the constant function $f\left(t\right)=1$defines an inner product on $C\left[a,b\right]$.

Substitute $\sqrt{{\int }_a^{b}w\left(t\right)f\left(t\right)f\left(t\right)dt}$for $\left(f,f\right)$into equation (2).

$$\begin{gathered} ||f||=\left(f,f\right) \\ =\sqrt{{\int }_a^{b}w\left(t\right)f\left(t\right)f\left(t\right)dt} \\ =\sqrt{{\int }_a^{b}w\left(t\right)dt} \\ =\sqrt{1} \end{gathered}$$

Solve further as:

$\left(f,f\right)=1$

Therefore, the value of the norm of the constant function $f\left(t\right)=1$defines an inner product on $C\left[a,b\right]$.