Inner Product Spaces 

Definition

MotivationSo far, we've seen that any two vectors can be naturally added and also multiplied by any scalar. But what if we want to define another structure for the vectors that gives a multiplicative result? The two vectors can be multiplied and this can be done by computing the dot product of the vectors. Equivalently, the dot product can also be defined as Dot product is used to find the angle between the two vectors. $\implies$ cos$\theta$ = $\dfrac{\vec{u}\cdot\vec{v}}{ \vec{u}   \vec{v} }$ Now suppose you want to calculate the work done in moving an object from one point to another, then we know that the work done is the product of force applied and the distance between two points ($W = Fd$) but this is only correct when the force is applied in the same direction in which the object is moving. Therefore, $W$ = $d$ $F$ cos$\theta$ which the dot product between the force and the displacement vector. 1 
Bird's Eye ViewOne of the very important applications of the dot product is finding projections. Projection of any vector $\vec{u}$ is its shadow on another vector $\vec{v}$ and is defined as a component ($\vec{u}$cos$\theta$). Mathematically, the projection of $\vec{u}$ onto $\vec{v}$ = $ \vec{v} $cos$\theta$ = $ \dfrac{\vec{u}\cdot\vec{v}}{ \vec{u} }$ where $\theta$ is the angle between $\vec{u}$ and $\vec{v}$. 3 
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Figure 1: Projection of a vector on another vector. 
The length of any vector $\vec{u}$ can be computed as: $ \vec{u} $ = $\sqrt{\vec{u}\cdot\vec{u}}$. For complex vectors, there is a generalized form of the dot product called Inner Product. The Inner Product of any two vectors $\vec{x}$ and $\vec{y}$ is defined as $\langle\vec{x}, \vec{y}\rangle$ = $\overline{\vec{x}}\cdot \vec{x}$, where $\vec{x}$ is the conjugate of $\vec{x}$. If we take any complex vector $\vec{x}$ then the length $ \vec{x} $ = $\sqrt{\langle\vec{x}, \vec{x}\rangle}$. Note: The conjugate of any complex number $z = a + ib$ is $\overline{z} = a  ib$. Now we can consider the above example where $\vec{u} = i$, then $ \vec{u} $ = $\sqrt{\langle\vec{x}, \vec{x}\rangle}$ = $\sqrt{\overline{ i }\cdot i}$ = $\sqrt{i\cdot i}$ = 1 Now, all the vectors in a space whose Inner Product can be computed forms an Inner Product Space.
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Context of the DefinitionWe often consider our vector spaces over a real field but here, we shall discuss the vector spaces over the complex field as well.
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Figure 2: Inner product of two vectors 
Let $V$ be a vector space over a field $F$, then the Inner Product defined is as $ \langle , \rangle : V$ x $V$ $\rightarrow$ $F$ is an operation that satisfies the following axioms: 3 
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Figure 3: Linearity of Inner Product 
3) <$\overline{u, v}$> = <$v, u$>, where <$\overline{u, v}$> represents the complex conjugate. 4) <$u, u$> $\geq$ 0 and <$u, u$> = 0 if and only if $u$ = 0. 5 
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Figure 4: Conjugate Symmetry and Positivity of Inner Product 
Lastly, the set ($V$, < , >) is called as Inner product space, where the Inner Product of two vectors is <$u, v$> = $\sum_{i=1}^{n} u_i v_i$. Example 1: Let $V$ = $\mathbb{R}^n$ and $u, v \in$ $V$ such that $u = \{u_1, u_2, . . . . . . . . u_n\}$ and $v = \{v_1, v_2, . . . . . . . v_n\}$ 1) Linearity Hence, $V$ = $\mathbb{R}^n$ forms an Inner Product space.
Example 2: Let $V = C [0, 2]$ be a vector space of all the continuous functions defined over real numbers. Hence, $V = C [0, 2]$ forms an Inner Product space. Note: The following animation uses the property of definite integrals, i.e., $\int_{a}^{b} f(x)$ $dx$ = Area under the curve. 9 
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Figure 5: Geometrical Illustration of Example2 
Norm of a vectorLet V be a vector space over a field F, then the length of vector $v \in$ V will be the Norm (denoted as $ v $), defined as Suppose V = $\mathbb{R}^2$ and the vector $v \in$ V.
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Figure 6: Interpretation of Norm as length 
Similarly, if we consider a cuboid then the length of its diagonal will be treated as the Norm of vector $v$ in $\mathbb{R}^3$. For a given scalar $c \in$ F,
We've seen so far that an Inner Product is all about the length and angle between the two vectors whereas a Normed vector space helps in determining a metric on a space that measures the distance between the two vectors. These spaces are ordered in such a way that, (Inner product spaces) $\subsetneq$ (Normed vector spaces) $\subsetneq$ (Metric spaces) $\subsetneq$ (Topological spaces) The Norm of any vector $\vec{u}$ is $ \vec{u} $ = $\sqrt{<\overline{u}, u>}$, we can say that the Inner Product is naturally associated with the Norm of vector and therefore, the Inner Product spaces can also be called a Normed vector spaces.
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Figure 7: Relation between Topological Metric, Normed and Inner Product Spaces 
Applications
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HistoryThroughout the nineteenth century, from the time of Fourier and Dirichlet, Sturm and Liouville itself the notions of “Vector Space” and “Inner Product” were emerged. It was Fourier who arose the idea of inner product to determine the coefficient of the Fourier Series. While the wave equation was being solved for the vibration of a string given an initial displacement, it was observed that the sin and cos functions were well suited for it. Clairaut and Euler then found the integral orthogonality conditions, which permitted them to figure out the coefficients in the Fourier series expansion of the data. It was however viewed that these conditions were consistent only for the initial data, as only around 17601800, the concept that every function could be expanded arose. 11 
Pause and Ponder1) If we define an Inner Product of two vectors in a vector space V = $\mathbb{C}$ as a classic dot product i.e $<u,v> = u\cdot v$, 2) Can you think of the possibilities where the Inner Product of the two vectors in a vector space be zero? 3) Let V be a vector space over some field F and $u, v, w \in$ V. Now if $<u, v> = <u, w>$, then what can be said about 11 
References[1] Lay, David C. Linear algebra and its applications. AddisonWesley,, 2012. [2] Birkhoff, G., & Kreyszig, E. (1984). The establishment of functional analysis. Historia Mathematica, 11(3), 258321. [3] levap (https://math.stackexchange.com/users/32262/levap), Relation between metric spaces, normed vector spaces, and inner product space., URL (version: 20180705): https://math.stackexchange.com/q/2841873 [4] Renze, John; Stover, Christopher; and Weisstein, Eric W. "Inner Product." From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/InnerProduct.html [5] http://www.princeton.edu/~aaa/Public/Teaching/ORF523/S17/ORF523_S17_Lec2_gh.pdf [6] https://prezi.com/p/2x9biebjhjwg/applicationsofinnerproductspaces/ [7] http://www.eng.fsu.edu/~dommelen/quantum/style_a/dot.html 12 
Further Reading[1] https://sites.math.washington.edu/~greenbau/Math_555/Course_Notes/555notes5.ps_pages.pdf [2] https://www.math.usm.edu/lambers/mat610/sum10/lecture2.pdf [3] https://www.math.tamu.edu/~yvorobet/MATH3042011A/Lect305web.pdf 13 
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