Questions tagged [inner-products]

For questions about inner products and inner product spaces, including questions about the dot product. An inner product space is a vector space equipped with an inner product. The dot product (seen in multivariable calculus and linear algebra) is a simple example of an inner product—other inner products may be seen as generalizations of the dot product.

Given vectors $x = (x_1, x_2, \dotsc, x_n)$ and $y = (y_1, y_2, \dotsc, y_n)$ in $\mathbb{R}^n$, the dot product of $x$ and $y$ is $$ x \cdot y = \sum_{j=1}^{n} x_j y_j. $$ The dot product on $\mathbb{R}^n$ is linear in both $x$ and $y$ and has the property that $x\cdot x \ge 0$ for all $x$, with equality if and only if $x = 0$. Moreover $x \cdot y = \lVert x\rVert \lVert y\rVert \cos(\theta)$, where $\lVert x\rVert$ denotes the length of $x$ and $\theta$ is the measure of the angle between the vectors $x$ and $y$. The dot product is then an algebraic tool which can be used to describe geometric properties of $\mathbb{R}^n$ (e.g. distance and angle).

An inner product is a generalization of the dot product. An inner product space is a vector space over a field $\mathbb K$ (either $\mathbb R$ or $\mathbb C$) endowed with a map $\langle\cdot,\cdot\rangle\colon V\times V\longrightarrow\mathbb K$ such that

  1. $(\forall v_1,v_2,v\in V):\langle v_1+v_2,v\rangle=\langle v_1,v\rangle+\langle v_2,v\rangle$;
  2. $(\forall v_1,v_2\in V)(\forall\lambda\in\mathbb{K}):\langle\lambda v_1,v_2\rangle=\lambda\langle v_1,v_2\rangle$;
  3. $(\forall v_1,v_2\in V):\langle v_1,v_2\rangle=\overline{\langle v_2,v_1\rangle}$;
  4. $(\forall v\in V):\langle v,v\rangle\geqslant0$ and $\langle v,v\rangle=0\iff v=0$.

Such a map is called an inner product. As an example, consider the space $\mathcal{C}\bigl([0,1]\bigr)$ of all continuous functions from $[0,1]$ into $\mathbb C$. If $f,g\in\mathcal{C}\bigl([0,1]\bigr)$, define$$\langle f,g\rangle=\int_0^1f(t)\overline{g(t)}\ \mathrm dt.$$

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Relationship between inner product and norm

I understand that there can be many different types of norms (e.g. mean norm, Cartesian norm, supremum norm etc). Are there also other types of inner products apart from $\langle x,y \rangle= \sum_{j =1}^n x_j y_j$? Also, I read that for any inner…
Student
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Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a triangle, we can see that the lengths of the third…
alok
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Why does cross product give a vector which is perpendicular to a plane

I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$ give me a vector which is perpendicular to a…
jjepsuomi
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What Is An Inner Product Space?

As I've understood it, what I've learned is that the dot product is just one of many possible inner product spaces. Can someone explain this concept? When is it useful to define it as something other than the dot product?
Emil H
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Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my textbook. I doubted about it: why do I have to…
Taxxi
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$\ell_p$ is Hilbert if and only if $p=2$

Can anybody please help me to prove this: Let $p$ be greater than or equal to $1$. Show that for the space $\ell_p=\{(u_n):\sum_{n=1}^\infty |u_n|^p<\infty\}$ of all $p$-summable sequences (with norm $||u||_p=\sqrt[p]{\sum_{n=1}^\infty |u_n|^p}\…
ccc
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An example of a norm which can't be generated by an inner product

I realize that every inner product defined on a vector space can give rise to a norm on that space. The converse, apparently is not true. I'd like an example of a norm which no inner product can generate.
luysii
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Intuition for the Cauchy-Schwarz inequality

I'm not looking for a mathematical proof; I'm looking for a visual one. I'm having trouble understanding (in my mind's eye) why the dot product of two vectors V and W produces a scalar that is less than the length of V multiplied by the length of…
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An orthonormal set cannot be a basis in an infinite dimension vector space?

I'm reading the Algebra book by Knapp and he mentions in passing that an orthonormal set in an infinite dimension vector space is "never large enough" to be a vector-space basis (i.e. that every vector can be written as a finite sum of vectors from…
Gadi A
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Understanding the Musical Isomorphisms in Vector Spaces

I am trying to solidify my understanding of the muscial isomorphisms in the context of vector spaces. I believe I understand the definitions but would appreciate corrections if my understanding is not correct. Also, as I have had some difficulty…
ItsNotObvious
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Why is orthogonal basis important?

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that special quality of orthogonal basis (extending to…
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Is complex conjugation needed for valid inner product?

What are the benefits of using a conjugate linear inner product in a complex vector space vs a simple linear inner product? That is, why do we demand that $(y,x) = \overline{(x,y)}$ as opposed to $(y,x)=(x,y)$? Of course, this ensures that $(x,x)$…
Obidiah
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How do you prove that $tr(B^{T} A )$ is a inner product?

Consider the vectorspace of all real $m \times n$ vectors and define an inner product $\langle A,B\rangle = \operatorname{tr}(B^T A)$. "tr" stands for "trace" which is the sum of the diagonal entries of a matrix. How do you prove…
Dieter Verbeemen
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What is "inner" about the inner product?

The inner product I am asking about is the one that generalizes the dot product for an arbitrary inner product space. Why is it called an "inner" product? Is there an outer product? Who named it so, and when?
MJD
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Inner product on $C(\mathbb R)$

With axiom of choice it is possible to construct an inner product on $C(\mathbb R)$. My question is, is it possible to explicitly construct an inner product on $C(\mathbb R)$? I.e. to give a closed formula to calculate the inner product? I know it…
daw
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