Polynomial and Functional Vector Spaces 

Definition
Let $V$ be a vector space over a field $F$ and $A$ be any set. If we define functions $f : A \rightarrow V$ and $g : A \rightarrow V$ and for any x $\in A, c \in F$, then
1) $(f + g)(x) = f(x) + g(x)$ 2) $(cf)(x) = cf(x)$ Let $P(\mathbb{R})$ be a set of all the polynomials with the coefficients from the field $\mathbb{R}$ and
$P_1(x), P_2(x) \in$ $P(\mathbb{R}$) then $P(\mathbb{R})$ will be a vector space if 1) $0 \in P(\mathbb{R}$) 2) $P_1 + P_2$ $\in P(\mathbb{R}$) 3) $c\cdot P_1$ $\in P(\mathbb{R}$), $\forall c \in \mathbb{R}$
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MotivationOne must have spent a considerable amount of time understanding what exactly a vector apace is before reaching at this point. We are also aware that a vector space can be something much more than just being a Euclidean space. It is something where we can perform linear algebraic operations: Addition and scalar multiplication and it can be done anywhere. Suppose we take any two shapes and combine them then the combined area will be the sum of their areas and the same goes with scalar multiplication. So we can see those shapes as vectors since they follow the vector space axioms. There are many more things other than the typically considered arrows and an array of numbers that can be modeled as vectors which we are unaware of. 0 
Bird's Eye ViewConsider a set of functions $F = \{ f(x)  \int_{a}^{b} f(x) dx \in \mathbb{R}\}$ 0 
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Figure 1: Integrable functions form a vector space. 
Context of the Definition
Note: $(f + g)(x)$ and $(cf)(x)$ are the notions for the new function obatined after adding and scaling the functions respectively. 0 
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Figure 2: Addition of two functions 
The above illustration works in the same manner as when we add the two vectors coordinate by coordinate in a Euclidean space. Just like them, we can also add the two functions and multiply them by a scalar to obtain another new function. 0 
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Figure 3: Scalar multiplication of a function 
Examples 1) If $V$ is a set of all the functions such that $f: \mathbb{R} \rightarrow \mathbb{R}$ then it forms a vector space over a real field.
A set of 'Polynomial Functions' is a Vector Space too! A Polynomial is also a kind of function whose variables have nonnegative exponents. We will briefly discuss here about a polynomial function behaving like a vector and the set of each one of them forms a vector space. It is also abundantly clear that the polynomials satisfy the distributive, associative, and commutative property. Consider a set of polynomial functions $P(\mathbb{R})$ with all the real coefficients from the field $\mathbb{R}$.
We've already seen that the zero polynomial exists in the set. Addition and scalar multiplication is defined as "componentwise" in P$(\mathbb{R})$. So, now we can conclude that the set of Polynomial functions is a vector space.
Finite and Infinite Dimensional Vector Spaces The vector space $V$ over a field $F$ is said to be a FiniteDimensional Vector Space if it is spanned by a finite set of vectors $\{ v_1, v_2, . . . . . . , v_k \}$. If $V$ cannot be spanned by a finite set of vectors, then $V$ is said to be an InfiniteDimensional Vector Space.
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ApplicationsFunction Spaces have played an important role in applied sciences as well as in mathematics itself. It was used for the development of the modern analysis of partial differential operators, distribution theory, numerical analysis, integral equations, approximation theory, and so on.

Pause and Ponder1) Let V be a vector space, W is a subspace of V and A be the set. 2) Will a set of all the differentiable real functions form a vector space? 0 
References[1] Mapa, S. K. (2003). Higher Algebra: Abstract And Linear (revised Ninth Edition). Sarat Book Distributors. [2] Gallian, J. (2012). Contemporary abstract algebra. Nelson Education. [3] Birkhoff, G., & Kreyszig, E. (1984). The establishment of functional analysis. Historia Mathematica, 11(3), 258321. [4] Sasane, Amol. "Functional analysis and its applications." [5] https://www.math.tamu.edu/~dallen/linear_algebra/chpt3.pdf [7] https://www.math.ubc.ca/~lior/teaching/1617/412_F16/Notes/InfiniteDimensions.pdf [8] http://www.math.lsa.umich.edu/~kesmith/infinite.pdf [9] https://dmpeli.math.mcmaster.ca/TeachProjects/Math1B03/Slides/lesson33.pdf 0 
Further Reading[1] https://dmpeli.math.mcmaster.ca/TeachProjects/Math1B03/Slides/lesson33.pdf [3] http://www.math.lsa.umich.edu/~kesmith/infinite.pdf 0 
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