Question

Let \(L^2[-1, 1] = \{f: [-1,1] \to \mathbb{R} : f \text{ is Lebesgue measurable and } \int_{-1}^1 |f(x)|^2dx < \infty\}\) and the norm \(||f||_2 = \left(\int_{-1}^1 |f(x)|^2 dx\right)^{1/2}\) for \(f \in L^2[-1,1]\).
Let \(F: (L^2[-1,1], ||.||_2) \to \mathbb{R}\) be defined by \[ F(f) = \int_{-1}^1 f(x)x^2dx \quad \text{for all } f \in L^2[-1,1]. \] If \(||F||\) denotes the norm of the linear functional F, then \(5||F||^2\) is equal to ................

Hint:

For any functional on \(L^2[a,b]\) of the form \(F(f) = \int_a^b f(x)g(x) dx\), the Riesz Representation Theorem immediately tells you that \(||F|| = ||g||_2\). The problem then reduces to calculating the norm of the known function \(g(x)\).

Answer 1

Step 1: Understanding the Concept:
The problem asks for the norm of a continuous linear functional on the Hilbert space \(L^2[-1,1]\). The Riesz Representation Theorem provides a powerful tool for this. It states that for any such functional \(F\), there exists a unique element \(g\) in the Hilbert space such that \(F(f) = \langle f, g \rangle\) for all \(f\), and the norm of the functional is equal to the norm of this element, i.e., \(||F|| = ||g||_2\).
Step 2: Key Formula or Approach:
1. Identify the inner product on \(L^2[-1,1]\): \(\langle f, g \rangle = \int_{-1}^1 f(x)g(x)dx\).
2. Compare the given functional \(F(f)\) with the inner product definition to find the representing function \(g(x)\).
3. Calculate the norm of \(g(x)\) using the \(L^2\) norm formula: \(||g||_2 = \left(\int_{-1}^1 |g(x)|^2 dx\right)^{1/2}\).
4. This gives \(||F||\). Then compute the final required value.
Step 3: Detailed Calculation:
The given functional is \(F(f) = \int_{-1}^1 f(x)x^2dx\).
The inner product in \(L^2[-1,1]\) is \(\langle f, g \rangle = \int_{-1}^1 f(x)g(x)dx\).
By comparing the form of \(F(f)\) with \(\langle f, g \rangle\), we can identify the representing function \(g(x)\) as: \[ g(x) = x^2 \] According to the Riesz Representation Theorem, the norm of the functional \(F\) is equal to the norm of \(g\): \[ ||F|| = ||g||_2 = ||x^2||_2 \] Now, we calculate \(||x^2||_2\). First, we find its square: \[ ||x^2||_2^2 = \int_{-1}^1 (x^2)^2 dx = \int_{-1}^1 x^4 dx \] Evaluate the integral: \[ \int_{-1}^1 x^4 dx = \left[ \frac{x^5}{5} \right]_{-1}^1 = \frac{(1)^5}{5} - \frac{(-1)^5}{5} = \frac{1}{5} - \left(-\frac{1}{5}\right) = \frac{2}{5} \] So, we have \(||F||^2 = \frac{2}{5}\).
The problem asks for the value of \(5||F||^2\). \[ 5||F||^2 = 5 \times \frac{2}{5} = 2 \] Step 4: Final Answer:
The value of \(5||F||^2\) is 2.
Step 5: Why This is Correct:
The solution correctly applies the Riesz Representation Theorem to find the norm of the functional by identifying the representing function \(g(x)=x^2\) and calculating its \(L^2\)-norm. The final arithmetic is straightforward.