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I am looking to describe all functions of type $f:\mathbb{R} \rightarrow \mathbb{R}$ as a vector space. Is this possible? What is a basis? Can I write any function of this type as $$g(x) = \int^{\infty}_{-\infty}a(k)sin(kx) + b(k)cos(kx)dk$$ for $a(k), b(k) \in \mathbb{R}$, and $k, \in \mathbb{R}$. I am assuming that $sin(kx)$ is a basis for this function space.

Let's suppose we have a basis of cardinality $Card(\mathbb{R})$, call it $B = \{b_1(kx), b_2(kx), \ldots \}, k \in \mathbb{R}$.

I don't understand why people are saying that the vector is a finite linear combination, but I can write a general function as

$$g(x) = \int^{\infty}_{-\infty}a_1(k)b_1(kx)+ a_2(k)b_2(kx) + \ldots dk$$

This seems like a continuous combination. Can someone explain this?

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  • $\begingroup$ Calling that integral $f(x)$, what is $f(-x)$? Can you write $\cos x$ as some integral of that type? $\endgroup$ – AHusain Aug 8 '18 at 16:21
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You can certainly define it as a vector space over the reals with pointwise addition as the addition of vectors, multiplication by reals as multiplication by elements of the field. You can verify all the axioms for a vector space. The additive identity is the zero function, inverses are the negative of the function, and so on.

Finding a basis is difficult. We only allow finite sums of basis vectors in a representation. The basis has to have the cardinality of the continuum.

You could also define the vector space that is spanned by the functions $\sin (kx)$. That is a fine vector space but it is missing many of the functions from $\Bbb R$ to $\Bbb R$. Because we only allow finite sums of basis elements, all the functions in this space are continuous.

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It depends on how you define “to describe”. Saying that it is the set of all functions from $\mathbb R$ into itself is a perfectly fine description. But don't expect to find an easy-to-describe basis of that space. And, no, the functions $x\mapsto\sin(kx)$ don't form a basis of this space. For instance, all those functions are odd, and therefore no even function (other than the null function) can be expressed as a linear combination of some of them.

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