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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Geometric interpretation for projection of a vector $x$ onto a subspace $U$.

There is a theorem that states: Let $u_1, \dots, u_n$ be an orthogonal basis for a subspace $U$ in an inner product space. The orthogonal projection of any vector $x$ onto $U$ is the point $\displaystyle p=\sum_{i=1}^{n}\left \langle x,\hat{u}_i…
Mathematicing
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Prove the properties of an inner product

$V$ is a $n$-dimensional Euclidean space with an inner product denoted by $(\quad,\quad)$. $\{{\alpha}_{1},{\alpha }_{2},\cdots,{\alpha}_{n}\}$ is a basis of $V.$ $({\alpha}_{i},{\alpha }_{j})\leq 0,i\ne j.$ Show that 1. For every nonzero vector…
user202406
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Prove that a Linear Transformation is Normal, such that $T^2=T$

Let V be a finite inner product space and $T:V\rightarrow V$ a linear transformation such that $$T^2=\frac{1}{2}(T+T^*)$$. Prove that $T$ is a "Normal Transformation", such that $$TT^*=T^*T$$ $T^2=T$ Proving the first statement isn't hard and I…
Alan
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Notion of weak convergence on a normed space without inner product

It seems weak convergence, $x_n \rightharpoonup x$, means that $\displaystyle\lim_{n\to\infty} \langle x_n,x\rangle = \langle x,x\rangle$. Now if we fix the left position of the inner product, then we have the functional $f_a(x) = \langle…
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Example of complete orthonormal set in an inner product space whose span is not dense

Let $X$ an inner product space and $A$ be an orthonormal set and $\overline{Span(A)}$ = $X$ then $A$ is Complete. But the converse is not true until we consider $X$ as a Hilbert space. I am searching of an example for that. Here, I call a system/set…
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Prove $(\sum\limits_k a_k b_k)^2 \leq \sum\limits_k b_k a_k^2 \sum\limits_k b_k$ using Cauchy Schwarz

Let $a,b$ be two $n$ dimensional vectors, we want to show that $(\sum\limits_k a_k b_k)^2 \leq \sum\limits_k b_k a_k^2 \sum\limits_k b_k$ Recall the Cauchy Schwarz inequality is given as $|\langle x,y \rangle| \leq \|x\|\|y\|$ then let $x = a,y =…
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Forming the orthogonal space is just a special case of forming a polar set?

Let $M\subset X$ be a subspace and define the polar set $M^{\circ}:=\{x^{\ast}\in X^{\ast}:|\langle u,x^{\ast}\rangle|\le 1\forall u\in M\}$ and $M^{\perp}:=\{x^{\ast}\in X^{\ast}:\langle u,x^{\ast}\rangle =0\forall u\in M\}$ I want to show…
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Do Algebraic Extensions have Inner Products?

Given that Algebraic Extensions are vector spaces over fields is it possible to define an inner product?
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Inner product on quantized enveloping algebra

I have a question about a procedure, described in section $2.1.5$ of "Quantum bounded symmetric domains". Here the author describes how to introduce an inner product on $U_q(\mathfrak{g})$. Therefore he uses the isomorphism belonging to the…
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Why is the inner product of a quadratic form a quadratic form?

I was going through a derivation of the second derivative of the $\log \det X$ where $X$ is symmetric positive definite, I noticed that despite the second order approximation of log det is written as: $f(Y) \approx f(X) + \langle \nabla f, x_o…
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what is the difference between a Hermitian inner product and an inner product?

Are there any difference? Or is the Hermitian inner product just a special case of an inner product on a complex space?
Mathematicing
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If $\mathrm{min}_{u\in U}||v-u||=||v-u_0||$, then $v-u_0\in U^{\perp}$

Problem Let $U$ be a subspace of an inner product space V (here V may not be finite dimensional). Fix $v\in V$. If there exists $u_0 \in U$ such that $\mathrm{min}_{u\in U}||v-u||=||v-u_0||$, then $v-u_0\in U^{\perp}$ Attempt If $t\in\mathbb{R}$…
illysial
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Dot Product Derviation

The dot product or inner product in Euclidean Space $A\cdot B$ has two definitions: Algebraically defined as: $$A \cdot B = \sum_{i=1}^{n}A_i \cdot B_i=A_1B_1 + A_2B_2 ... A_nB_n$$ Geometrically defined as: $$A \cdot B = \Vert A \Vert \Vert B \Vert…
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Matrix tensor indices

Suppose I have an orthonormal basis $$B = \left \{ u_{i} \right \}_{i=1}^{\infty}$$ Then for a matrix $K$, do I represent it as $$K = \sum_{j,k=0}^{\infty}k_{jk}\left ( u_{i}\bigotimes u_{j} \right )$$ or $$K = \sum_{j,k=0}^{\infty}k_{jk}\left (…
Mathematicing
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Are there any books/papers talking about inner product on vectors over finite fields?

Are there any books/papers talking about inner product on vectors over finite fields? In particular, I'd like to learn things on $F_p^n$, or simply $F_2^n$. I read some proofs using the inner product on $F_2^n$ in some papers, but I can only…
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