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I am currently reading "Applied Analysis", which could be found here,

and on page 85 I don't understand example 4.16. There it is said:

Suppose that $X$ is the space of all real-valued functions on the interval $[a,b]$. We may identify a function $f: [a,b] \to \mathbb{R}$ with a point $\prod_{x\in [a,b]} f(x)$ in $\mathbb{R}^{[a,b]}$, so $X = \mathbb{R}^{[a,b]}$ is the $[a,b]$-fold Cartesian product of $\mathbb{R}$.

In this construction the expression $\prod_{x\in [a,b]} f(x)$ should almost always yield $\infty$ because its a product over an uncountable set and so this identification is not one-to-one, or does i read $\prod_{x\in [a,b]} f(x)$ wrong?

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3 Answers 3

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You’re reading $\prod_{x\in[a,b]}f(x)$ incorrectly. (This is perfectly understandable, because the notation is very poor.) It does not mean a product of the real numbers $f(x)$ for $x\in[a,b]$.

Let $I$ be an index set, and for each $i\in I$ let $Y_i$ be a set. Then $$X=\prod_{i\in I}Y_i\;,$$ the Cartesian product of the sets $Y_i$, is defined to be the set of all functions $$x:I\to\bigcup_{i\in I}Y_i$$ such that $x(i)\in Y_i$ for each $i\in I$. Each of these functions is a single point of the product set $X$. I generally write $x=\langle x(i):i\in I\rangle$ to describe such a point.

In your setting $I=[a,b]$, and $Y_i=\Bbb R$ for each $i\in[a,b]$. Thus, in this case $x=\langle x(i):i\in I\rangle\in X$ is actually a function from $I=[a,b]$ to $\Bbb R$, since each of the sets $Y_i$ is just $\Bbb R$. Change my $x$ to $f$ and substitute the specific index set for $I$, and you have a point $f=\langle f(x):x\in[a,b]\rangle\in X$. The author of that book has ill-advisedly chosen to write this as if it were a product of numbers, which is ridiculous, but his $\prod_{x\in[a,b]}f(x)$ should be understood exactly as my $f=\langle f(x):x\in[a,b]\rangle$.

He could have been a little less wrong by writing $\prod_{x\in[a,b]}\{f(x)\}$, a Cartesian product of singleton sets; that at least could not be interpreted as a product of numbers, but it also wouldn’t be a point in $X$: rather, it would be a singleton subset of $X$, a set with only one point in it.

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As there already are answers addressing the notation issue, I will just add some examples and intuition.

Consider a function $f : \{1,2,3,\ldots,n\} \to \mathbb{R}$. However, there is a bijection between a set of such functions and the $\mathbb{R}^n$. Take an $n$-dimensional vector $v \in \mathbb{R}^n$ such that $v = \langle v_1, v_2, v_3, \ldots, v_n \rangle$ and $v_k = f(k)$, and you could see that $\Phi(f) = v$ could be an example. Of course, if you have a point in $u \in \mathbb{R}^n$, then you could easily reconstruct the function: $g(k) = u_k$. The fact that the space of the functions is denoted as $\mathbb{R}^{\{1,2,\ldots,n\}}$ is not a coincidence.

This example can be generalized, e.g. take $f : \mathbb{N} \to \mathbb{R}$. Every such function is a point in the space of infinite sequences of real numbers, e.g. $(a_n)_{n \in \mathbb{N}}$ where $a_n = f(n)$. On the other hand, the space of such sequence can be thought of as an infinite Cartesian product of $\mathbb{R}$, and indeed it is somtimes denoted as $\mathbb{R}^\infty$.

We can go with this even further and further, e.g. $g : [a,b] \to \mathbb{R}$ is a point in space $\mathbb{R}^{[a,b]}$ that can be understood as infinite Cartesian product, but with uncountable number of coordinates it ranges over.

Please note, that with the above in mind, the sums (i.e. $\sum_{k=1}^n \cdot$) and series (i.e. $\sum_{k=1}^\infty \cdot$) are nothing else than the integrals (i.e. $\int_{\Omega}\cdot$) over discrete spaces. This is also one of the reasons why people tend to think of integrals as generalized sums.

Hope this helps ;-)

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How do you read $A^2$? It is all the functions from $\{0,1\}$ into $A$, but this is also the set of ordered pairs, or $2$-tuples if you will.

We can think of the ordered pair $\langle a,b\rangle$ as the function $f(0)=a, f(1)=b$.

Similarly we can consider other sets, larger sets, and generalize this: $\mathbb R^{[a,b]}$ is the set of all functions whose domain is $[a,b]$ and codomain $\mathbb R$. How do we identify a function $f$? We identify it with the sequence, or $[a,b]$-tuple in an extreme abuse of language, $\langle f(x)\mid x\in[a,b]\rangle$.

The product, is not a product of numbers but rather a product of sets, in our case a product of singletons. Note that we can write $f=\prod\limits_{x\in[a,b]}\{f(x)\}$. The problem begins when abusing the notation and identifying a point with its singleton, then $\prod\limits_{x\in[a,b]}\{f(x)\}$ turns into $\prod\limits_{x\in[a,b]}f(x)$ which may seem as a product of numbers instead of product of sets.

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