The dimension is the number of coordinates needed to specify a point on the object. For example, a rectangle is two-dimensional, while a cube is three-dimensional. The dimension of an object is sometimes also called its dimensionality , Really i want to know the geometric interpretation of infinit dimension as example we take the field $\mathbb{R}$ .

Question What is the geometric interpretation of $\mathbb{R}^{\infty}$ or function defined as infinit space vector by $f:\mathbb{R}\to \mathbb{R}$ ? .

Note: I meant to deal with function as vector .

Edit: I edited the question to clarify what I meant by $\mathbb{R}^{\infty}$

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    $\begingroup$ You can think of it as a set that consists of all possible infinite sequences of real numbers. In formal definition: $\mathbb{R}^\infty=\{(x_1,x_2,\ldots)\colon x_1,x_2,\ldots \in \mathbb{R}\}$. $\endgroup$ – Scippy Feb 12 '17 at 23:28
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    $\begingroup$ There are different kinds of infinity. The set of functions $f:\mathbb{R}\to\mathbb{R}$ could be denoted that way, too. Better to use $\mathbb{R}^{\mathbb{R}}$ or $\mathbb{R}^{\mathbb{N}}$ to be clear. $\endgroup$ – Jack D'Aurizio Feb 12 '17 at 23:32

As Jack D'Aurizio wrote, you have to be specific about what kind of infinity you're talking about. The easiest to imagine is probably countable infinity. For a finite vector you are probably used to writing $x=(x_1,x_2,x_3)$. You can also imagine this as a function, though:

$$x:\{1,2,3\}\to\mathbb R\qquad i\mapsto x_i$$

So in a sense, $x$ is a function from a finite set of indices to the set of real numbers, returning the coordinate for each index. This generalizes easily to $\mathbb R^{\mathbb N}$:

$$x:\mathbb N\to\mathbb R\qquad i\mapsto x_i$$

Now you have a countably infinite set of indices, and for each index you have a real coordinate. But while you are at it, if you think of a vector more like a function, then you can just as well write

$$x:\mathbb R\to\mathbb R\qquad i\mapsto x(i)$$

and suddenly you have $\mathbb R^{\mathbb R}$, a vector whose dimension is the cardinality of the reals. Treating functions as vectors is probably the most important application of infinite-dimensional vector spaces.

So as you asked about geometric interpretation, I'd think about functions with a geometric aspect to them. One typical example of an infinite-dimensional vector space would be the sine and cosine waves you use as base functions for a Fourier transformation. Then each Fourier coefficient is a essentially coordinate in the corresponding vector, with these waves as the basis. Perhaps others can come up with even more geometric applications.


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