Definition in Lax "sequence of continuous functions tending to $\delta$", are distributions needed for understanding?

Question

I'm trying to read Lax's functional analysis. In chapter 11 he makes a definition which I don't like.

A sequence of continuous functions ${k_n}$ on a $[-1,1]$ tends to $\delta$ if $\int_{-1}^{1} f(x)k_n(x)dx=f(0)$ for all $f \in C[-1,1]$

Within a proof where he establishes iff correspondence between this definition being valid for some ${k_n}$ and something similar to ${k_n}$ begin good kernels he refers to weak* convergence within the dual of $C[-1,1]$ which he declared to be the space of finite signed measures a few pages earlier. He says that the above statement with the integral is equal to $ $w*$ - \lim k_n= \delta$

I don't see how this is trivial. In what way or in which framework is the "definition" and its relation to weak* convergence clearer? Do I have to go all the way into distributions?

Answer 1

I'd say his definition doesn't define what $\delta$ is, only what "converging to $\delta$" means.

Just like in analysis, you say the sequence $a_n$ is said to converge to $+\infty$ if $$\forall M \in \mathbb{R},\ \exists N \in \mathbb{N},\ \forall n > N,\ a_n > M$$

This doesn't define what $+\infty$ is.

Now he can prove that this definition of "converging to $\delta$" is the same as weak* convergence to the linear form $\delta$ defined by $$\delta(f) = f(0)$$ But notice that his first definition doesn't define what $\delta$ is

Answer 2

Since $\delta$ is a distribution, the shorter answer is, yes you do need distribution theory to understand this.

However, if you are only concerned with the identity if $\int_{-1}^1 f(x)k_n(x) dx = f(0)$ for $f \in C([-1,1])$, then the right hand side is just $\delta(f)$.

On the left hand side, you probably have to use the definition of $k_n$, change of variables and the Lebesgue's Dominated Convergence Theorem.

Answer 3

You don't really need distribution theory to see this, but some topology.

The weak*-topology of $C[-1,1]$ is the topology induced by the maps $\Gamma_f: C^*[-1,1] \to \Bbb{R}$, $\Gamma_f(\Lambda) = \Lambda(f)$. Your book is claiming that this topology is the topology of pointwise convergence, meaning that $\Gamma_i \to \Gamma$ in $C^*$ if and only if $\Gamma_i(f) \to \Gamma(f)$ for all $f \in C$.

To see this, assume first that $(\Lambda_i) \subset C^*[-1,1]$ is a sequence such that $\Lambda_i(\varphi) \to \Lambda (\varphi)$ for all $\varphi \in C[-1,1]$, where $\Lambda \in C^*[-1,1]$. Let $B$ be an element of the local basis for the weak*-topology of $C^*$. This means that \begin{align} B = \bigcap_{k=1}^n \Gamma_{\varphi_k}^{-1} V_k = \bigcap_{k=1}^n \{ \Lambda \in C^*[-1,1] : \Lambda (\varphi_k) \in V_k\}\,, \end{align} where $V_k \subset \Bbb{R}$ are neighborhoods of the origin. Ṇow, because $\Lambda_i (\varphi) \to \Lambda (\varphi)$ for all $\varphi$, $\Lambda_i \varphi_k - \Lambda \varphi_k = (\Lambda_i - \Lambda)(\varphi) \in V_k$ for big enough $i$. So for all elements $B$ of the local basis we have $\Lambda_i \in \Lambda + B$ once $i$ is big enough. This means that $\Lambda_i \varphi \to \Lambda \varphi$ with respect to the topology of $C^*[-1,1]$.

Conversely, if $\Lambda_i \to \Lambda$ with respect to the weak*-topology, it follows that for all $(\varphi_n)_{n=1}^m$ and $(r_i)_{n=1}^m$ there exists a natural number $k$ such that \begin{align} i \geq k & \implies \Lambda_i \in \Lambda + \{ \Lambda \in C^*[-1,1] : | \Gamma_{\varphi_n}(\Lambda)| < r_n, 1 \leq n \leq m\} \\ & \iff |\Lambda_i \varphi_n - \Lambda \varphi_n| < r_n\, \quad 1 \leq n \leq m\,. \end{align} This is pointwise convergence once we choose $m = 1$, so $\Lambda_i \varphi \to \Lambda \varphi$ pointwise.