Question

We say two norms \(\|u\|_{1}\) and \(\|u\|_{2}\) on a vector space \(V\) are equivalent if there exist constants \(A\) and \(B\) such that $$ \|u\|_{1} \leq A\|u\|_{2} \quad \text { and }\|u\|_{2} \leq B\|u\|_{1} $$ for all \(u \in V\). If two norms are equivalent then show the following: (a) If \(u_{n} \rightarrow u\) with respect to one norm then this is also true for the other norm. (b) Every linear functional that is continuous with respect to one norm is continuous with respect to the other norm. (c) Let \(V=C[0,1]\) be the vector space of continuous complex functions on the interval \([0,1]\). By considering the sequence of functions $$ f_{n}(x)=\frac{n}{\sqrt{\pi}} \mathrm{e}^{-m^{2} x^{2}} $$ show that the norms $$ \|f\|_{1}=\sqrt{\int_{0}^{1}|f|^{2} \mathrm{~d} x} \text { and }\|f\|_{2}=\max \\{f(x)|| 0 \leq x \leq 1\\} $$ are not equivalent. (d) Show that the linear functional defined by \(F(f)=f(1)\) is contunuous wath respect to \(\|\cdot\|_{2}\) but not with respect to \(\|\cdot\|_{1}\).

Short answer

The conditions for norm equivalence: each sequence that converges in one norm also converges in another, and every linear functional, continuous in one norm, is also continuous in the second, are confirmed for norms \(\|u\|_{1}\) and \(\|u\|_{2}\). On the other hand, for the given functions \(f_n(x)\), norms \(\|f\|_{1}\) and \(\|f\|_{2}\) are not equivalent as their limits differ for \(n \rightarrow \infty\). The functional \(F(f) = f(1)\) is continuous with respect to the maximum norm \(\|f\|_{2}\), but not with respect to the integral norm \(\|f\|_{1}\).

Step-by-step solution

Part (a) Proof

Given that the sequence \(u_n\) converges to \(u\) with respect to norm1 (\(\|u\|_{1}\)), this means there exists a positive integer \(N\) such that for all \(n \geq N\), \[\|u_n - u\|_{1} < \varepsilon\]Take \(A = \|u_n - u\|_{1}\) and \(B= \|u_n - u\|_{2}\). Since \(\|u\|_{1} \leq A\|u\|_{2}\), it follows that \(B \leq A\|u_n - u\|_{2}\), which in turn implies that \(\|u_n - u\|_{2} < \varepsilon\). Hence, the convergence of \(u_n\) to \(u\) in norm1 implies convergence in norm2.

Part (b) Proof

Assume \(L: V \rightarrow W\) is a linear functional that is continuous with respect to norm1 (\(\|u\|_{1}\)). This means that, for every \(u, v \in V\) and \(\varepsilon > 0\), there exists \(\delta > 0\) such that, if \(\|u - v\|_{1} < \delta\), then \(\|L(u) - L(v)\| < \varepsilon\). Knowing that \(\|u\|_{1} \leq A\|u\|_{2}\), we can deduce \(\|u - v\|_{1} < A\|u - v\|_{2} < \delta\), yielding \(\|L(u) - L(v)\| < \varepsilon\). This shows that \(L\) is continuous with respect to norm2 (\(\|u\|_{2}\)).

Part (c) Proof

For \(n \rightarrow \infty\), \(f_n(x)\) approaches infinity at \(x = 0\) which implies that \(\|f_n\|_{2} \rightarrow \infty\). However, the integrand of the formula for \(\|f_n\|_{1}\) is \(f_n^2(x)e^{-2n^2x^2}\), and this tends to zero as \(n \rightarrow \infty\). As a result, \(\|f_n\|_{1}\) also tends to zero. This proves that the two norms \(\|f_n\|_{1}\) and \(\|f_n\|_{2}\) are not equivalent as one goes to infinity while the other approaches zero.

Part (d) Proof

We know that the functional \(F(f) = f(1)\) is continuous with respect to \(\|f\|_{2}\) because \(|F(f_n) - F(f)| = |f_n(1) - f(1)| \leq \|f_n - f\|_{2}\), which tends to zero as \(n \rightarrow \infty\). However, with respect to \(\|f\|_{1}\), we cannot guarantee that \(|F(f_n) - F(f)|\) tends to zero because the integrand of the formula for \(\|f_n - f\|_{1}\) may not tend to zero, so \(F(f)\) is not continuous with respect to \(\|f\|_{1}\).

Key concepts

Convergence in Normed Spaces

When we talk about convergence in normed spaces, we're referring to how a sequence of vectors behaves under a certain norm. If a sequence \(u_n\) converges to a vector \(u\), it means that as you go further in the sequence, the vectors get closer to \(u\), according to the norm used. Essentially, the distance between \(u_n\) and \(u\) becomes smaller and smaller.

Norm equivalence is key here. When two norms \(\|u\|_1\) and \(\|u\|_2\) are equivalent, convergence in one norm implies convergence in the other. This is because the norms are "comparable" through constants \(A\) and \(B\), allowing us to say if \(u_n\) converges to \(u\) under one norm, it will do so in the other.

**Key Takeaways:**

• Convergence in normed spaces deals with how sequences of vectors behave or approach a limit.
• Equivalent norms assure that convergence in one norm implies convergence in another, smoothing the transition between different mathematical approaches.

Linear Functionals Continuity

A linear functional is a function defined on a vector space that preserves addition and scalar multiplication. When we say a linear functional is continuous, it means small changes in the input result in small changes in the output.

With equivalent norms, if a linear functional is continuous with respect to one norm, it must be continuous with respect to the other. This is due to the bounded difference between the two norms, allowing the linear functional to behave similarly across both norms.

**Understanding Continuity:**

• Linear functionals map vectors to real numbers or other scalars.
• Continuity in this context relies on the differences in input yielding small changes in output, a property retained through equivalent norms.
• Maintaining continuity across norms aids in the stability of mathematical models and solutions.

Equivalent Norms

Equivalent norms are a powerful concept in linear algebra and functional analysis. Two norms \(\|u\|_1\) and \(\|u\|_2\) are considered equivalent if there are constants \(A\) and \(B\) such that for all vectors \(u\), \(\|u\|_1 \leq A\|u\|_2\) and \(\|u\|_2 \leq B\|u\|_1\). This means they essentially measure size in a way that is mutually compatible.

In practical terms, this equivalence means that the properties of vectors (like convergence and continuity) will be preserved irrespective of which norm is used. However, not all norms turn out to be equivalent.

**Highlights of Equivalent Norms:**

• Defines a relationship where two norms are scalable by fixed constants.
• Preserves major vector space properties across different norms for consistency.
• Not every pair of norms are equivalent, as seen in the space of continuous functions.

Space of Continuous Functions

The space of continuous functions, often denoted \(C[0,1]\), consists of all functions that are continuous on the interval \([0,1]\). A unique feature of this space is that functions have no discontinuities, and both pointwise evaluation and integration can be used to assess them.

In this space, we can define various norms, like the \(L^2\) norm (an integral form) and the supremum norm (based on the maximum value). Notably, these norms are not always equivalent. For instance, the example given shows a sequence of functions where one norm tends to zero while the other tends to infinity. This demonstrates the nonequivalence of norms in this space.

**Key Insights about Continuous Functions:**

• Functions are smooth without breaks across an interval.
• Different norms can measure different attributes like size or spread of the function.
• Understanding nonequivalence of norms helps in knowing the limitations of each measure in function spaces.