An inner product space is a vector space equipped with an inner product. The inner product is a generalisation 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 ...
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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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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 ...
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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 ...
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How to interpret the adjoint?

Let $V \neq \{\mathbf{0}\}$ be a inner product space, and let $f:V \to V$ be a linear transformation on $V$. I understand the definition1 of the adjoint of $f$ (denoted by $f^*$), but I can't say I ...
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Maximizing a sum of inner products

Someone asked this question on a French maths forum here and it caught my attention. The question is the following: let $(E, \langle \cdot, \cdot \rangle)$ be a Euclidean vector space. Find the ...
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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 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 ...
14
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Multiplicative norm on $\mathbb{R}[X]$.

How to prove that : there is no function $N\colon \mathbb{R}[X] \rightarrow \mathbb{R}$, such that : $N$ is a norm of $\mathbb{R}$-vector space and $N(PQ)=N(P)N(Q)$ for all $P,Q \in \mathbb{R}[X]$. ...
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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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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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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 $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
12
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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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What is a complex inner product space “really”?

To be clear on this, I know what is the definition of an inner product space and some properties and theorems about them. What I am asking for is an intuition for this definition in the complex case. ...
12
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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 ...
12
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In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?

$(V,\|\cdot|)$ be a real normed linear space such that $\|x\|=\|y\|$ implies $\lim\limits_{n\to\infty} \|x+ny\|-\|nx+y\|=0$, then is it true that the norm comes from an inner-product space ?
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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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Motive for the definition of inner product

Mathematicians pride themselves on writing proofs of propositions in an elegant way, but frequently (maybe even usually?) neglect to formally write motivations of definitions with the same elegance, ...
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480 views

How to motivate the axioms for the inner product

Typically, one doesn't just write down lists of axioms and then sees if there are enough interesting examples that satisfy them; they evolve over time, usually from a couple of very ...
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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 ...
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Is Cross Product Defined on Vector Space?

In Wikipedia, a cross product between two "vectors" is defined in terms of the angle between the vectors and their magnitudes. As I learned cross product in linear algebra, which I understand to be ...
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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 does orthogonality mean in function space?

The functions $x$ and $x^2 - {1\over2}$ are orthogonal with respect to their inner product on the interval [0, 1]. However, when you graph the two functions, they do not look orthogonal at all. So ...
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Show that if $T_1$, $T_2$ are normal operators that commutes then $T_1+T_2$ and $T_1T_2$ are normal.

Let $V$ be a finite dimensional inner-product space, and suppose that $T_1$, $T_2$ are normal operators on $V$ that commutes. How to show that $T_1+T_2$ and $T_1T_2$ are then normal? It is clear if ...
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Equivalent inner products on a Hilbert space

Take a Hilbert space $(\mathcal H,(\cdot,\cdot)_{\mathcal H})$ and two equivalent inner products $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$ on $\mathcal H$, i.e. such that there are $a,b \in \mathbb R$ ...
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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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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?
8
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Inner product space over $\mathbb{R}$

Definition of the problem I have to prove the following statement: Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an inner product space over $\mathbb{R}$. prove that for all $x,y\in ...
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Alternating two-form

Let $V$ be a real vector space of dimension $2$, and let $\langle\ \ ,\ \ \rangle$ be an inner product on $V$. Define $f:V^4 \to \mathbb{R}$ by $$f(x,y,z,w):=\langle x,y \rangle \langle z,w \rangle- ...
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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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Then prove: $\vec{v} = \vec{0}$ if $\langle u,v\rangle = 0$

If $\vec{v} \in V$ such that $\langle u,v\rangle = 0$, $\forall \vec{u} \in V$. Then prove: $\vec{v} = \vec{0}$ I tired to solve by assuming that they are $\langle u,v\rangle \neq 0$ $\rightarrow$ ...
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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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A specific type of orthogonal bases

In $\mathbb{R}^n$, can we find an orthogonal basis $v_1,v_2,\ldots,v_n$ with $\|v_i\|=1$ such that $$\|\bar{v}_1\|=\|\bar{v}_2\|= \cdots =\|\bar{v}_n\|$$ where $\bar{v}_i\in\mathbb{R}^m$ is the vector ...
7
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Is a bra the adjoint of a ket?

The instructor in my quantum computation course sometimes uses the equivalence $$(\left|a\right>)^\dagger\equiv\left<a\right|$$ I understand that this is true for the typical matrix ...
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Proving: $2n (\sum_{i=1}^{2n} a_i^2) \geq (\sum_{i=1}^{2n} a_i)^2+(\sum_{i=1}^{2n} a_i (-1)^i)^2$

Let $a_i\in \mathbb{R}$ and $n$ an integer. How do you prove: $$2n \left(\sum_{i=1}^{2n} a_i^2\right) \geq \left(\sum_{i=1}^{2n} a_i\right)^2+\left(\sum_{i=1}^{2n} a_i (-1)^i\right)^2?$$ This is ...
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A Banach Manifold with a Riemannian Metric?

Given an infinite dimensional manifold modeled on a Banach space, what does it mean for it to have a Riemannian metric? Does it necessarily mean that it is actually a Hilbert manifold? My ...
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Energy estimate of the differential equation $\dot{x}=Ax$

Conside the differential equation $$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...
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Topological space $\nRightarrow$ Metric space $\nRightarrow$ Normed space $\nRightarrow$ Inner product space (Examples)

If I have an inner product space, the hierarchy goes: Inner product space $\Rightarrow$ normed space $\Rightarrow$ metric space $\Rightarrow$ topological space. The reverse, however, is not always ...
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Does this inner product on $L^1([0,1])$ have a name?

Math people: For $f, g \in L^1([0,1])$, define $$\langle f,g \rangle = \int_0^1 \int_0^1 f(t)g(t')\exp(-|t-t'|)dt'\,dt.$$ Although we don't normally think of $L^1([0,1])$ as an inner product space, ...
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Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
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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 ...
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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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Isomorphisms of inner-product spaces

I think I understand why all finite-dimensional vector spaces over a field $\mathbb{K}$ are isomorphic to $\mathbb{K}^n$. Any linear map $T: V \rightarrow W$ between finite-dimensional vector spaces ...
6
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Cauchy-Schwarz for metrics with arbitrary signatures

When the norm of a vector is always greater than or equal to zero, the Cauchy-Schwarz inequality holds, but what if we look at a metric with an arbitrary signature? Then the inner product of a vector ...
6
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Show that $V = U^\perp \bigoplus U$

If $(V,\langle , \rangle)$ is a Euclidean vector space, $U \subseteq V$ is a subspace of V and $U^\perp := \{v \in V | \langle v,u \rangle = 0, \forall u \in U\}$. Show $V = U^\perp \bigoplus U$ In ...
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Rotations and reflections in ${\bf R}^3$.

By a rotation in ${\bf R}^3$ I mean an orthogonal linear transformation $f:{\bf R}^3\to {\bf R}^3$ represented by a matrix $A$ (i.e. $fx=Ax$) with $\det A=1$. By a reflection (through $S$) I mean an ...
6
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Why isn't it a Hilbert space

Let $X$ be the vector space of all the continuous complex-valued functions on $[0,1]$. Then $X$ has an inner product $$(f,g) = \int_0^1 f(t)\overline{g(t)} dt$$ to make it an inner product space. But ...
6
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Given a vector space with two inner products, there is a linear transformation taking one to another

I am looking for some hint to the following question: Let $V$ be an $n$-dimensional real inner product space and let $\langle x,y\rangle$ and $[x,y] $ both be two different inner products on V. ...
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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). ...