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It is well - known that a point - wise multiplication operation can be defined on some function spaces.

For example, $C([0,1])$, the vector space of a real - valued (say) continuous functions $f$ defined on the unit - interval $[0,1] \subset \mathbb{R}$, admits such a multiplication, defined as \begin{equation} (fg)(x) := f(x)g(x), \quad f,g \in C([0,1]) \end{equation}

Now, for function spaces whose elements are defined on a countable set, say \begin{equation} \mathbb{R}^k = \{ x: \mathbb{N}_k \to \mathbb{R}, \quad n \mapsto x_n, \quad n = 1, \dots, k \} \quad (k \in \mathbb{N}) \end{equation} or \begin{equation} \mathbb{R}^\infty = \{ x: \mathbb{N} \to \mathbb{R}, \quad n \mapsto x_n, \quad n = 1,2 \dots \,\} \quad \end{equation}

I wonder why this is not done analogously. So, define pointwise multiplication by \begin{equation} (xy)(n) := x_ny_n \end{equation}

For example, in the "column - notation", two elements of $\mathbb{R}^2$, $x = (x_1,x_2)$ and $y = (y_1,y_2)$ would then have the product $ xy = (x_1y_1, x_2y_2)$.

I realize there must be something that makes this obvious operation utterly useless, for otherwise, it would be used commonly. Any hints as to why this attempt to define pointwise multiplication on these functions spaces is uninteresting would be great !

From what I understand so far (unfortunately far to little), I wonder whether the cardinality of the domain (uncountable in the case $[0,1]$, countable in the cases $\mathbb{N}_k = \{1,\dots k\}$ and $\mathbb{N}$) makes a crucial difference, whether there is some algebraic property that breaks down, or whether it is something completely different ?

Thanks for your feedback and help!

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Note that pointwise multiplication in a function space is induced by a notion of multiplication in the range space. Also, each example you gave is induced by pointwise multiplication in the space $X=C(\{1,\ldots, k\};\mathbb R)$ when $C(\mathbb R^k)$ is treated as $C(X)$. –  Alex Becker Apr 13 '12 at 23:08
    
Also cardinality makes no difference. Your definition can be extended to arbitrary sets $S$ by consider the range to be the algebra $X=F(S,\mathbb R)$ of functions from $S$ to $\mathbb R$ with pointwise addition and multiplication. –  Alex Becker Apr 13 '12 at 23:24
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Why do you think it must be "utterly useless" simply because applications of it happen to be rare? Why can't there be things that are "only occassionally useful" without being either "utterly useless" or "used commonly"? –  Henning Makholm Apr 13 '12 at 23:29
    
@Henning hm - you're absolutely right there is no reason to devalue structeres that do not have an immediate application! So, provided my definition above makes sense, I didn't intend to say it would be uninteresting to study it for its own sake. –  harlekin Apr 13 '12 at 23:51
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up vote 2 down vote accepted

Your last example is the Hadamard product $x\circ y$; $x,y\in{\mathbb R}^{n}$, which is defined for matrices. However, it is an overkill and very difficult to work with if you talk about (column) vectors. Instead, you can use an equivalent form $x\circ y={\rm diag}(x)y$, where ${\rm diag}(x)\in{\mathbb R}^{n\times n}$ is a diagonal matrix with vector $x$ on its main diagonal, and the matrix-vector product ${\rm diag}(x)y$ is understood in the usual linear algebraic sense. This form is handy when one of the vectors is known ($x$ in this case) and the other is to be found. If both vectors are unknown, then the 'outer' product $xy^{T}$ may be helpful. Your pointwise product can be extracted from the diagonal of this rank-one matrix, i.e., $x\circ y={\rm diag}(xy^{T})$.

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There is nothing inherently uninteresting about pointwise multiplication. If you have a vector space of functions which is closed under pointwise multiplication, it is an algebra. If the multiplication behaves nicely with respect to a norm or other structures, you may have a Banach algebra or a $C^*$-algebra. These structures are well studied and can be found in many textbooks, and indeed, $C([0,1])$ is an example of a $C^*$-algebra.

On the other hand, there are many interesting function spaces which are not closed under pointwise multiplication. $L^p$ spaces are probably the most obvious example: the product of two functions from $L^p$ need not be in $L^p$. So it is still useful to develop theory for function spaces without reference to the pointwise multiplication.

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