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Questions tagged [inner-product-space]

An inner product space is a vector space equipped with an inner product. The inner product is a generalization of the “dot” product often used in vector calculus.

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Norms Induced by Inner Products and the Parallelogram Law

Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...
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Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric space} &...
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Is there a vector space that cannot be an inner product space?

Quick question: Can I define some inner product on any arbitrary vector space such that it becomes an inner product space? If yes, how can I prove this? If no, what would be a counter example? Thanks ...
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For complex matrices, if $\langle Ax,x\rangle=\langle Bx,x\rangle$ for all $x$, then $\langle Ax,y\rangle=\langle Bx,y\rangle$ for all $x$ and $y$?

Given $A$ and $B$, $n\times n$ complex matrices. If $\langle x,y\rangle =y^{*}x$ for all $x,y\in \mathbb C^{n}$, then the following are equivalent: (1) $\langle Ax,y\rangle=\langle Bx,y\rangle$, for ...
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How to understand dot product is the angle's cosine?

How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized) Thinking about how to prove this in the most intuitive way resulted in proving a ...
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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 ...
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A counterexample to theorem about orthogonal projection

Can someone give me an example of noncomplete inner product space $H$, its closed linear subspace of $H_0$ and element $x\in H$ such that there is no orthogonal projection of $x$ on $H_0$. In other ...
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Inner Product Spaces over Finite Fields

Inner product spaces are defined over a field $\mathbb{F}$ which is either $\mathbb{R}$ or $\mathbb{C}$. I want to know what happens if we try to define them over some finite field. Here's an example:...
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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 $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
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1answer
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What is the geometric meaning of the inner product of two functions?

When it comes to inner product I have thus far only dealt with vectors, and so the concept is very intuitive because one can easily visualize two vectors and how they get multiplied, and it is clear ...
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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 ...
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Isometry group of a norm is always contained in some Isometry group of an inner product?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $Let $||\cdot||$ be a norm on a finite dimensional real vector space $V$. Does there always exist some inner product $\<,\>$ on $V$ such ...
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What is the general formula for calculating dot and cross products in spherical coordinates?

I was writing a C++ class for working with 3D vectors. I have written operations in the Cartesian coordinates easily, but I'm stuck and very confused at spherical coordinates. I googled my question ...
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1answer
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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 ...
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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 ...
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1answer
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Vector Algebra Coordinate Transformation

Let us look at two coordinate systems $K$ and $K'$ with axes, respectively, $(x_1,x_2,x_3)$ and $(x_1',x_2',x_3')$ and unit vectors ($\vec{e_1},\vec{e_2},\vec{e_3}$) and ($\vec{e_1'},\vec{e_2'},\vec{...
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Cauchy-Schwarz inequality proof (but not the usual one)

Before you downvote/vote-to-close, I am not asking for a proof of: $$\sum^n_{i=1}a_ib_i\le\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum^n_{i=1}b_i^2 } $$ Which is what EVERY link I've found assumes is the ...
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Characterisation of inner products preserved by an automorphism

Let $V$ be a finite dimensional vector space. Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry. (You can ...
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Differentiating an Inner Product

If $(V, \langle \cdot, \cdot \rangle)$ is a finite-dimensional inner product space and $f,g : \mathbb{R} \longrightarrow V$ are differentiable functions, a straightforward calculation with components ...
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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 ...
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Is Inner product continuous when one arg is fixed?

In a inner product space with inner product $\langle\ ,\ \rangle$ and real or complex line as its base field, for each point $x$ in the space, is $\langle x,-\rangle$ continuous function on the second ...
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Projection of $z$ onto the affine set $\{x\mid Ax = b\}$

Suppose $A$ is fat(number of columns > number of rows) and full row rank. The projection of $z$ onto $\{x\mid Ax = b\}$ is (affine) $$P(z) = z - A^T(AA^T)^{-1}(Az-b)$$ How to show this? ...
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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 ...
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A map which commutes with Hodge dual is conformal?

$\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\id}{\operatorname{Id}}$ Let $V$ and $W$ be $d$-dimensional, ...
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Derivation of the polarization identities?

For a real (or complex) inner product space $V$, the inner product can be expressed in terms of the norm as either $$ \langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2) $$ or $$ \langle x,y\rangle=\...
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1answer
124 views

A norm which is symmetric enough is induced by an inner product?

$\newcommand{\<}{\langle} \newcommand{\>}{\rangle} $ It is a fact that for every norm $\| \|$ on a finite dimensional (real) vector space, its isometry group $\text{ISO}(|| \cdot ||)$ is ...
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1answer
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Show $(W_1 + W_2 )^\perp = W_1^\perp \cap W_2^\perp$ and $W_1^\perp + W_2^\perp ⊆ (W_1 \cap W_2 )^\perp$

Given $V$ a inner product space and $W_1$, $W_2$ subspaces of $V$ Show $(W_1 + W_2 )^\perp = W_1^\perp \cap W_2^\perp$ and $W_1^\perp + W_2^\perp ⊆ (W_1 \cap W_2 )^\perp$.
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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 ...
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Distance of a matrix from the orthogonal group

Let $\lVert A\rVert = \left(\sum_{i,j=1}^n \left\| a_{ij} \right\|^2\right)^{\frac{1}{2}} = \sqrt{\operatorname{tr}(A^\top A)}$ be the Frobenius norm on $n \times n$ matrices. Fix $A \in GL_n(\mathbb{...
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1answer
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Proving that if $\langle Ax,x\rangle =0$ for every $x$, then $A$ is the zero operator

I feel kind of dumb but I needed this little claim as a part of a proof I'm writing, and I figured out that I'd better just ask, since I could not find the correct algebraic manipulation needed in ...
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1answer
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Continuity of scalar product

In a Hilbert space $H$ with inner product and associated norm, why would if $\|x-x_n\| \longrightarrow 0$ and $\|y-y_n\| \longrightarrow 0$ also $\langle x_n,y_n\rangle \longrightarrow\langle x,y\...
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1answer
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Do any books or articles develop basic Euclidean geometry from the perspective of “inner product affine spaces”?

Definitions. By a vector space, I simply mean an $\mathbb{R}$-module. By an affine space, I mean a vector space $X$ (the "translation space") together with a set $P$ (of "points"), together with an ...
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In a complex vector space, $\langle Tx,x \rangle=0 \implies T = 0$

Suppose $T$ is a linear operator on a complex inner product space. Is it a theorem that if $\langle Tx,x\rangle=0$ for all $x$ in the space then $T=0$. The theorem fails in the real case, as seen for ...
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Rigid motion on $\mathbb{R}^2$ which fixes the origin is linear

Let $V=\mathbb{R}^2$ be an inner product space with the standard inner product, and let $T$ be a rigid motion of $V$. Suppose $T(0)=0$, prove that $T$ is linear. (A rigid motion of an inner product ...
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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,...
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4answers
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difference between dot product and inner product

I was wondering if a dot product is technically a term used when discussing the product of $2$ vectors is equal to $0$. And would anyone agree that an inner product is a term used when discussing the ...
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1answer
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Difference between an isometric operator and a unitary operator on a Hilbert space

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is an identity operator, $^*$ is a binary operation.) What ...
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Proofs of the Cauchy-Schwarz Inequality?

How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?
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2answers
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Is a norm a continuous function?

Is a norm on a set a continuous function with respect to the topology induced by the norm? Is a topology on the set that can make the norm continuous (i.e. the topology that is compatible with the ...
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6answers
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Prove that the eigenvalues of a real symmetric matrix are real.

I am having a difficult time with the following question. Any help will be much appreciated. Let $A$ be an $n×n$ real matrix such that $A^T = A$. We call such matrices “symmetric.” Prove that the ...
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A Banach space that is not a Hilbert space

Can someone give me an example of a Banach space that is not a Hilbert space? I can't think of any because I don't know how to show one space that can not have inner product structure.
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Under what conditions is a linear automorphism an isometry of some inner product?

Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and that $T: V \to V$ is a (linear) isomorphism. When is it possible to construct an inner product on $V$ making $T$ an ...
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1answer
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Norm with symmetric positive definite matrix

If B is $n \times n$ real symmetric positive definite matrix, then $(x,y) = x^T By$ defines an inner product on $R^n$. How do you prove that $\|x\|=(x^T B x)^{1/2}$ is a norm on $R^n$?
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1answer
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$l^p$ space not having inner product

I know that $l^2$ space is a Hilbert space. But for other $l^p$ spaces, where $p\geq1$, I have to show that they do not satisfy the parallelogram equality. But, I can't find appropriate sequences ...
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2answers
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TVS: Topology vs. Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
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3answers
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How to prove that $L^p [0,1]$ isn't induced by an inner product? for $p\neq 2$

I'd like to know how could I prove that $L^p [0,1]$ isn't induced by an inner product? (For $p\neq 2$, including $p=\inf$). It is clear to me that I would need to find two functions $f$, $g$ in $L^p$ ...
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Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$

Let $V$ be a real inner product space and $S=\{v_1,v_2, \dots, v_n\}\subset V$. How am I to prove that $S$ is linearly independent if and only if the determinant of the matrix $$ a_{ij}=\pmatrix{\...
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$T^*T=TT^*$ and $T^2=T$. Prove $T$ is self adjoint: $T=T^*$ [duplicate]

$V$ is an inner product space of finite dimension over $\mathbb{R}$, and $T:V\to V$ a linear transformation which is normal, that is, $T^*T=TT^*$. In addition $T^2=T$. Prove $T$ is self adjoint, that ...
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If the unit sphere of a normed space is homogeneous is the space an inner product space?

Consider a normed vector space $V$. Suppose that for every pair of unit vectors $v,w$ there exists a linear isometry which sends $v$ to $w$ (and leaves the subspace spanned by $v$ and $w$ invariant). ...
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A subspace $X$ is closed iff $X =( X^\perp)^\perp$

Let $X$ be a subspace of a Hilbert space $H$. prove: $X$ is closed iff $X =( X^\perp)^\perp$ I do not know how to proceed. any hints would be appreciated. thanks. $X^\perp = ${ $y \in H$:$\...