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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Are two hilbert spaces with the same algebraic dimension (their hamel bases have the same cardinality) isomorphic?

We know that two hilbert spaces tat have orthonormal bases of the same cardinality are isomorphic (as an inner product spaces). my question is what can we say when we know that their hamel bases ...
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“Normalizing” a Function

One of our homework problems for my linear algebra course this week is: On $\mathcal{P}_2(\mathbb{R})$, consider the inner product given by $$ \left<p, q\right> = \int_0^1 p(x)q(x) \, dx $$ ...
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Continuous real valued functions and inner product space?

Let $V$ be the space of all continuous real valued functions on the interval $[1,4]$ with the inner product defined by: $$\langle f,g\rangle = \int_1^{4} f(t)g(t)\,dt.$$ (i) Find an ...
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225 views

Orthogonality of Complex Vectors Under the Euclidean Dot Product

We all have a good interpretation of orthogonal vectors in Euclidean Space, how does this extend to Complex vectors? Algebraically, the thing that bugs me, is that the dot product is only symmetric up ...
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Proving that $|\langle x,y\rangle|=\|x\|\|y\|$ iff $x,y$ are linearly dependent

Show: For any vectors $x$ and $y$ in an inner product space $V$ over $\mathbb{F}$, we have $$\left|\langle x,y\rangle\right|=\|x\|\|y\| \iff x,y \ are \ dependent$$ ($\leftarrow$) If $x$ and $y$ ...
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Inner Product on Complex Vector Space

Let $\left< . , .\right>$ denote an inner product on $V$ where $V \subseteq \mathbb{C}^n$. I'm having trouble understanding why $$ \left< u, iv\right>i = \left< u, v\right> $$ ...
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38 views

$\left< u, v\right> \stackrel{?}{=} \left< v, u \right>$

For some vector space $V$, is it true that $$ \left< u, v \right> = \left< v, u \right> $$ for all $u, v \in V$? Does this only hold if $V \subseteq \mathbb{R}^n$ or if $V \subseteq ...
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A problem on an operator in inner product space

Let $T \in L(V,W)$ . Define $T^* : W \to V$ by $$\langle T^*w,v\rangle = \langle w,Tv\rangle$$ is it true that $\dim \operatorname{Range}(T^*) = \dim \operatorname{Range}(T)$ ? Here all the vector ...
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60 views

Can square root of a normal operator be normal too ??

Suppose $T$ is normal on a complex inner product space, I have proved that $T$ has a square root $S$. Can it be normal too ?
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inequality related to transformations and inner products

Let $T$ be a bounded transformation from a hilbert space to itself. Suppose that if $||f||\leq 1$ and $||g||<1$ then $|\text{Re}(Tf,g)|\leq M$ where we are taking the real part of the inner ...
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From a Hilbert space to another Hilber space, is a norm preserving invertible transformation also unitary?

Given two Hilber spaces - $H_1, H_2$ and a transformation $T:H_1 \to H_2$ that is norm preserving and invertable, does this imply that $T$ is also unitary transformation, namely that it preserves the ...
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Indices Contraction in Minkowski Spacetime

Why is it that $$\partial_\mu\partial^\mu=\partial_t^2-\nabla^2$$ (this I believe is called the D'Alembert operator.) but $$\partial_\mu j^\mu=\dot{j^0}+\nabla\cdot \vec j?$$ Why is there a minus on ...
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35 views

is it true that $Range(T) \subset Range(T^*)$ always?

is it true that $Range(T) \subset Range(T^*)$ always? or, in some special case. How to prove ? Only a hint is enough.
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Historically first axiomatization of the inner product

When was the dot/scalar/inner (however you want to call it) product historically first abstractly introduced as a mapping that had to fulfill certain axioms ?
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96 views

A basic question on adjoint operator

Suppose we have a non-zero vector $v$ for which $Tv=0$. Then can we say that there exist a non-zero vector $w$ such that $T^*w=0$ where $T^*$ is adjoint operator.
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Dot product over complex vectors: Conjugate first or second?

Does there exist a truly "standard" dot product over complex vectors? Wikipedia and Wolfram's MathWorld indicate directly or indirectly that the second argument is conjugated. Matlab's ...
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Why is the inner product not an element of the Hilbert space?

What I know about Hilbert space is that, elements in that space can be complex numbers. But I was confused to read this statement from a book: The inner product, being a complex number, is not an ...
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Determine whether $\langle f, g\rangle = \int_1^e {1 \over x} f(x)g(x)\,dx $ is a inner product.

Let $C[1,e]$ be the set of continuous real-valued functions with domain $W:=[1,e]$. Let $$\langle f, g\rangle = \int_1^e {1 \over x} f(x)g(x)\,dx $$ be a function. Determine whether $\langle \cdot, ...
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Flat space Minkowski metric

I am having some problem understanding the why in Minkowski spacetime, the continuity equation is written as $$\partial_\mu J^\mu=0.....................(*)$$ Physically, I know that $$\partial_t ...
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Inner Product Space on linear transformation on itself

So $V$ is an inner product space and $T : V \to V$ is a linear map such that $$||T(v)|| = ||v||$$ for all $v \in V$. Prove that $$\langle T(v), T(w)\rangle = \langle v, w\rangle$$ for all $v,w \in V$. ...
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Is every normed vector space, an inner product space

Let $V$ be a vector space over $\mathbb{C}$. If $V$ is an inner product space, then $V$ is normed (where the norm is defined as $\|x\|=\sqrt{(x,x)}\,\,$). Now if $V$ is normed, does it follow that ...
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$\max_{1 \leq i \leq n}|\langle x,w_i\rangle|$, $\max_{1 \leq i \leq n}|\langle y,w_i\rangle|$ at same $w_i$ if $x$ and $y$ are close enough?

Let $x,y \in \mathbb{C}^n$ with $|x|_2 = |y|_2 = 1$. Let $w_1, \ldots, w_N \in \mathbb{C}^n$. Let $j,k \in \{1,\ldots, N\}$ such that $$ |\langle x,w_j\rangle| = \max_{1 \leq i \leq n}|\langle ...
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Linear Algebra: Distance between two parallel lines

Find the distance between the two (obviously parallel) lines below where $\alpha ,\beta \in \mathbb R$ are scalars. $$\text{Line ...
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A basic question on linear maps in inner product space

Suppose we have a linear map $S:V \to V$ with the property that $\langle Sv,v\rangle =0$ for all $v \in V$. Then is it true that $\langle Su,v \rangle =\langle Sv,u\rangle$ for any $u,v \in V$?
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What is the dot product between a vector of matrices?

There is a notation used in many sources (e.g. Wikipedia: http://en.wikipedia.org/wiki/Exponential_family) for the natural parameters of exponential family distributions which I do not understand, and ...
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A basic question on an unique vector identifying a linear functional in an inner product space

Let $f:V \to C$ be a non-zero linear functional where $dim(V)=l-1$. Write $V= kerf \oplus span\{v_0\}$ where $v_0 \perp ker f$. Now I am able to prove that there exists a multiple of $v_0$ (call it ...
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85 views

Cauchy Schwarz inequality true with positive semi-definite inner product space

In one of exercises in a linear algebra book I have been asked to prove the following : "Suppose we modify the inner product definition such that $\langle u,u\rangle=0$ need not imply $u=0$." I have ...
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Why orthogonal basis?

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 ...
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Derivative of inner product (taking limit inside)

For each $x \in [a,b]$ let $A_x: H \to H$ be an operator on a Hilbert space. The inner product $(A_xu,v)_{H}$ can be thought of as a function from $[a,b] \to \mathbb{R}.$ I want to say that ...
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Dot product of the column vectors from a matrix and their transposes through matrix multiplication

I have a matrix with data, every dataset is a column vector in my matrix. I want to know the dot product of the transpose of each column vector with the original column vector. If I transpose the ...
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64 views

Inner Product Space

Let $X$ be the set of all $n$-tuples of complex numbers and let $x,y \in X$ such that $x=(a_1,a_2,\dots,a_n)$ and $y=(b_1,b_2,\dots,b_n)$. The inner product of the two vectors is given by $\langle ...
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Representation of Inner Products ( Inner Product Matrix).

I am struggling with understanding this: For an inner product of $\mathbb{R}^3$ defined by $\langle x,y\rangle = 2x_1y_1 -x_1y_2 -x_2y_1 + 5x_2y_2$ the matrix relative to the standard basis is:- ...
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Prove a definition function is inner product

I want prove the formula below be to an inner product: Given two non-zero vectors $u$ and $v$, with $A$ symmetric and positive definite, $$\langle u,v\rangle_A:=\langle Au,v\rangle=\langle ...
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Choosing sets of vectors on a complex sphere

Consider a complex $t$ dimensional unit sphere. Can we have $t$ sets of $2^t$ vectors $v_{ij}\in \Bbb C^t$ on the sphere where $i=1$ to $t$ and $j=1$ to $2^t$ on this with inner products satisfying ...
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Help me go from English to Logic

The positive-definiteness axiom used for just about all the definitions of inner-product spaces that I've seen goes like this: $$\langle \mathbf{x},\mathbf{x}\rangle \ge 0 \text{ with equality only ...
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Hoffman & Kunze exercise

This is a problem from Hoffman & Kunze book on linear algebra: Let V be a finite dimensional inner-product vector space. Let U be a self-adjoint unitary linear operator over V. Show that ...
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Inner product space proof

How do we prove that ||v|| = ||w|| iff (v-w) $ \bot $ (v+w). I proceeded by starting from $ \langle v-w,v+w \rangle $ =0 and reach at $\langle v,v\rangle + \langle v,w \rangle $$= \langle w,v ...
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Orthogonality and inner product space

The question relates to orthogonality and inner product spaces. If $ W,W_{1}, W_{2} $ are subspaces of finite dimensional vector space V. We have to prove the following (a) $ W_{1} \subset W_{2} ...
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Why doesn't Cauchy-Schwarz in $\mathbb{R}^n$ generalize to exponents $k>2$?

Given $(x_i)_{i=1}^n, (y_i)_{i=1}^n \in \mathbb{R}^n$, the Cauchy-Schwarz Inequality asserts $$\left( \sum_{i=1}^n x_i y_i \right)^2 \leq \left( \sum_{i=1}^n x_i \right)^2 \left( \sum_{i=1}^n y_i ...
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Show that $u_0=u_1$

$u_0$ and $u_1$ are vectors in the inner product space V with $\langle u_0,v\rangle=\langle u_1,v\rangle$ for all $v\in V$. Show that $u_0=u_1$ (let $v=u_0-u_1$). I'm would be glad to have a little ...
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A problem with linear operator in a Hilbert space

Let $(H,(\cdot,\cdot)_H)$ and $(Q,(\cdot,\cdot)_Q)$ two Hilbert separable spaces s.t $H\subset Q$ and let $B:H\to Q$ a bounded and linear operator. Let $\sigma,\tau\in H$ two fixed elements. My ...
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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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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 ...
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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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85 views

A problem on the bounds of Lp-norms

Let $L>0$ and $\Omega$ be the set of all integrable functions from $[0,L]$ to $[0,+\infty]$. Also, Let $f\in \Omega$ such that $\left \| f \right \|_{1}=1$. Find the tightest possible bounds for: ...
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A basic question on orthogonal vector

Let $V$ be a finite dimensional vector space and $X$ be a subspace. Let $$\langle u,y\rangle=0 \forall u $$ with the property that $$\langle u,x\rangle =0 \;\forall x \in X$$ where $u,y \in V$. Then ...
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Find the fallacy in using the Cauchy–Schwarz inequality

Let $\int_{a}^{b}\frac{f(x)}{x}dx=k$, wherein $f(x),a,b,k$ are positive. According to the Cauchy–Schwarz inequality: $\int_{a}^{b}xf(x)dx=\int_{a}^{b}x^{2}\frac{f(x)}{x}dx\leq \left ( ...
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Prove or disprove this argument

Let $L>0$ and let $\Omega$ be the set of all integrable functions from $[0,L]$ to $]0,+\infty[$. For all $\varphi, \psi \in \Omega$ define $\left \langle \varphi,\psi \right ...
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188 views

Are there any connections between metric space and inner product space

As mentioned in title, are the any connections between inner product space(well, here we talk about only real space) and metric space? I kind of notice that the axioms satisfied by both inner product ...
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Conditions of a constant inner product?

For a defined $f$, define $g$ such that: $\langle f,g \rangle = \int_a^b f(x)g(x)\,dx =cte \neq 0$