Prove Dual Space Isometry: X = C[0,1] Sup Norm

[complexnumber]

Homework Statement

For the Banach space $$X = C[0,1]$$ with the supremum norm, fix
an element $$g \in X$$ and define a map $$\varphi_g : X \to \mathbb{C}$$
by
$$\begin{align*}<br /> \varphi_g(h) := \int^1_0 g(t) h(t) dt, \qquad h \in X<br /> \end{align*}$$
Define $$W := \{ \varphi_g | g \in X \}$$.


Prove that $$\varphi_g \in X^*$$ and calculate
$$||\varphi_g||_{X^*}$$.

Homework Equations

The Attempt at a Solution

The supremum norm for a function $$f \in C[0,1]$$ is $$\displaystyle<br /> ||f|| = \sup_{x \in [0,1]} |f(x)|$$.

For $$h_1,h_2 \in X$$ and $$\lambda \in \mathbb{C}$$
$$\begin{align*}<br /> \varphi_g(h_1 + h_2) =& \int^1_0 g(t) (h_1(t) + h_2(t)) dt =<br /> \int^1_0 g(t) h_1(t) dt + \int^1_0 g(t) h_2(t) dt = \varphi_g(h_1) +<br /> \varphi_g(h_2) \\<br /> \varphi_g(\lambda h_1) =& \int^1_0 \lambda g(t) h_1(t) dt = \lambda<br /> \int^1_0 g(t) h_1(t) dt<br /> \end{align*}$$
So $$\varphi_g$$ is linear functional.

For any $$h \in X$$, $$\varphi_g$$ is continuous if $$\forall \varepsilon<br /> > 0$$ $$\exists \delta > 0$$ such that any $$h' \in X$$ satisfies
$$\begin{align*}<br /> d(h,h') < \delta \implies d(\varphi_g(h), \varphi_g(h')) <<br /> \varepsilon \text{.}<br /> \end{align*}$$
The distance function is $$\displaystyle d(h,h') = ||h - h'|| =<br /> \sup_{x \in [0,1]} |h(t) - h'(t)|$$.
$$\begin{align*}<br /> d(\varphi_g(h), \varphi_g(h')) =& |\int^1_0 g(t) h(t) dt -<br /> \int^1_0 g(t) h'(t) dt | = |\int^1_0 g(t) h(t) - g(t) h'(t) dt | \\<br /> =& |\int^1_0 g(t) (h(t) - h'(t)) dt| \leq \int^1_0 |g(t) (h(t) - h'(t))| dt \\<br /> \leq & \int^1_0 |g(t)| |(h(t) - h'(t))| dt \leq \sup_{t \in<br /> [0,1]} |(h(t) - h'(t))| \int^1_0 |g(t)| dt \\<br /> =& d(h,h') \int^1_0 |g(t)| dt<br /> \end{align*}$$
For any $$\varepsilon > 0$$, we get $$\displaystyle d(h,h') \int^1_0<br /> |g(t)| dt < \varepsilon$$ when $$\displaystyle d(h,h') <<br /> \frac{\varepsilon}{\displaystyle \int^1_0 |g(t)| dt} = \delta$$.
Thus $$\varphi_g$$ is continuous. Are the above proofs correct?

The formula for $$||\varphi_g||_{X^*}$$ is
$$\begin{align*}<br /> ||\varphi_g||_{X^*} :=& \sup \{ |\varphi_g(h)| | \norm{h} = 1<br /> \} \\<br /> =& \sup \left\{ |\int^1_0 g(t)h(t) dt| \bigg| \sup_{t \in [0,1]}<br /> |h| = 1 \right\}<br /> \end{align*}$$
How can I find the value of this?

 

[Office_Shredder]

For the supremum, maybe some sort of triangle inequality application can give you a starting upper bound

 

[kompik]

Are the above proofs correct?

As far as I can say, they are.

The formula for $$||\varphi_g||_{X^*}$$ is
$$\begin{align*}<br /> ||\varphi_g||_{X^*} :=& \sup \{ |\varphi_g(h)| | \norm{h} = 1<br /> \} \\<br /> =& \sup \left\{ |\int^1_0 g(t)h(t) dt| \bigg| \sup_{t \in [0,1]}<br /> |h| = 1 \right\}<br /> \end{align*}$$
How can I find the value of this?

You have
$$\left|\int^1_0 g(t)h(t) dt\right| \le \int^1_0 |g(t)| |h(t)| dt \le \int^1_0 |g(t)| dt$$
therefore
$$||\varphi_g|| \le \int^1_0 |g(t)| dt$$.

I think that $$||\varphi_g|| = \int^1_0 |g(t)| dt$$ holds. A naive approach to show this would be trying to approximate the step function $$h(t)=g(t)/|g(t)|$$ by continuous functions; but I guess there is a simpler solution.

BTW the same question was asked here: http://www.mathhelpforum.com/math-h...tial-geometry/146627-dual-space-isometry.html

 

[complexnumber]

Is this correct?

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