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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Finding an orthonormal basis for the space $P_2$ with respect to a given inner product

I am so confused on what to do for this question. The questions asks to find an orthonormal basis of $P_2$, the space of quadratic polynomials, with respect to the inner product $$ \langle p, ...
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If $\|u\| \leq \|u+av\|$ for all $a \in F$, How can I show that $\langle u,v\rangle=0$?

If $\|u\| \leq \|u+av\|$ for all $a \in F$, How can I show that $\langle u,v\rangle=0$? I know a standard solution uses $\operatorname{Re}$ and $t =$ something but was wondering if there was ...
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show that $\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' (t)\rangle$

Let $\gamma,\eta:[a,b]\to \mathbb R^n$ be continuous, differentiable, curves. show that $$\frac{d}{dt}\langle \gamma(t),\eta(t)\rangle =\langle\gamma '(t),\eta (t)\rangle +\langle \gamma (t),\eta ' ...
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How do I find the Jacobi matrix?

I've never done questions like these, so I would very much like some help. We are given a function $f: \mathbb R^n \to \mathbb R$ given by $f(x)=\langle x,\xi\rangle^2$ where $\langle\,,\rangle$ is ...
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The space of continuous functions $C([0,1])$ is not complete in the $L^2$ norm

I am trying to prove that the Euclidean Norm/inner product on $C([0,1])$ does not give rise to a complete metric space. To do this I am trying to find a Cauchy Sequence which does not converge in ...
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inner product space definition

I have some problem in the definition of inner product space. The book I use to learn in linear algebra and its application 4th edition (David C.Lay) In the chapter 6.7 it define the inner product ...
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Finding orthogonal complement

Let $X$ be an inner product space and let $x\in X$. $M=\{z\in X:\langle z,x\rangle=0\}$. I want to find $M^{\perp}$ and $M^{\perp \perp}$. Clearly, $\{x\}\subset M^\perp$. Thus $M^{\perp ...
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Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?

Quick question about subspaces, just to make sure I have this straight in my head. Taking an $n\times k$ matrix X with $rank(X)=k$, is every vector in $\mathbb{R}^n$ in either the column space $C(X)$ ...
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Inner-product question

Let $V$ be $\mathbf{R}^2$ equipped with usual inner product, and $v$ be a nonzero vector. $S_v(u)= u- 2 \frac{\langle u,v\rangle}{\langle v,v\rangle } v$ and $\Phi$ be a non-empty set of unit vectors ...
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Equivalent definitions of isometry

Consider a map $T:\mathbb{R}^2\to\mathbb{R}^2$ such that $\lVert T(x)\rVert=\lVert x\rVert$. Is this equivalent to stating that $\langle x, y\rangle=\langle T(x), T(y)\rangle$ for all ...
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Prove that orthogonality in Euclidean space is geometrically perpendicularity?

Is this simply true by definition (that is, taken as axioms?) How would one to prove that for $||\vec{x}||=1$ and $||\vec{y}||=1$, if $(\vec{x},\vec{y})=0$, then $\vec{x}\perp\vec{y}$? In other ...
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$||u||\leq ||u+av|| \Longrightarrow \langle u,v\rangle=0$

Prove that $\langle u,v\rangle=0\Longleftrightarrow ||u||\leq ||u+av||$. So far I can get the $\Longrightarrow$ very easily, but I need some help with the $\Longleftarrow$ implication, any hints ...
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Continuity of a multivariate function

I'm trying to show that $\langle , \rangle$ is continuous on $V{\times}V$, ($V$ an inner product space). I've tried approaching it by showing $\langle\vec x,\vec y\rangle\rightarrow\langle\vec a,\vec ...
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Necessity of completeness of the inner product space in Riesz representation theorem

I wanted to find a counter example to show that the completeness of the inner product space is necessary in Riesz representation theorem. Please give an example of a bounded linear functional $T$ on ...
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Show that $T\neq{T^*}$

Let $V=P_2(\mathbb{R}), T\in \mathcal{L}(P_2(\mathbb{R})),$ where $T(p)=(a_1x)$. Make $V$ an inner product space by defining $$\langle p,q\rangle=\int_0^1{p(x)q(x)\,dx}$$ So I calculate $$\langle ...
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find the distance between 2 functions

f(x) = $e^x$. g(x) = $.5(e-1/e) + 3x/e$. How do you find||f-g||. The inner product is defined as $\int_{-1}^1 f(x)g(x) dx$. I've tried this: $\int_{-1}^1 (e^x - (.5(e- e^{-1} + 3x/e)))^2dx$. This ...
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Prove $\langle x,x \rangle < 0$ or $\langle x,x \rangle > 0$ for all $x \neq 0$

[Added by PLC: This question is a followup to this already answered question.] Keep the axioms for a real inner product (symmetry, linearity, and homogeneity). But make the fourth be $$\langle x,x ...
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Homework: Second derivative of $\langle Ax, x \rangle$

So let $A \in M_{n}$ and define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(x) = \langle Ax, x \rangle $. Find f' and f''. After some work, I found the first derivative to be $f'(x)(v) = \langle Ax, v ...
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Distribution Property of Dot Product

Since the dot product has the property that for three vectors $a,b,c$ $a \cdot (b+c) = a \cdot b + a \cdot c$ Is that also true for $(a+b) \cdot (a+b) = a\cdot a + 2a\cdot b + b\cdot b$ ? Thank ...
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For inner product spaces, do we have $||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||$?

Let $V$ be an inner product space. Then for all $\vec{u},\vec{v} \in V$ we have $$||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||.$$ I know that the converse to the equation is true such that ...
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Show a linear transform is self adjoint - check my answer

We are given $T:V \to V$ a normal linear transform (meaning $TT^*=T^*T$) We are also given $T^2=T$. Show that $T$ is self adjoint (meaning $T^*=T$). What I did I think I may have done something ...
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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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Inner product question

We are given an inner product of $\mathbb R^3$: $f\left(\begin{pmatrix} x_1\\x_2\\x_3\end{pmatrix},\begin{pmatrix} y_1\\y_2\\y_3\end{pmatrix}\right) = ...
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Is any norm induced by some inner product? [duplicate]

It is a well-know fact that an inner product induces some norm. How about the converse? I think it's false but I can't think of an example. I'm thinking some properties like the parallelogram law ...
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Matrix Composed of Traces of Linear Independent Set of Matrices

I got the following problem: Let $S=\{A_1,A_2,...,A_k\} \subseteq \mathbb{M^R}_{n\times n}$ be a linear independent set of $k$ real $n \times n$ matrices with respect to the standard matrix inner ...
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Prove that set is orthonormal set

In vector space of all real polynomials with inner product $(x,y) = \sum_0^1x(t)y(t)dt$. $x_n(t) = t^n$ for $n = 0, 1, \dots $. Show that functions: $y_0(t) = 1, $ $y_1(t) = \sqrt(3)(2t-1), $ ...
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Prove series converges absolutely

Dot product is defined $\sum_{n=1}^\infty x_ny_n$. Let V consist of all sequences for which $\sum_{n=1}^\infty x_n^2$ converges. Prove that this series converges absolutely. I know I need to use the ...
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$T: \mathbb R^n \to \mathbb R^n$, $\langle Tu,v\rangle=\langle u,T^*v\rangle$, is $T^*=T^t$ regardless of inner product?

Basic question in linear algebra here. $T$ is a linear transform from $\mathbb R^n$ to $\mathbb R^n$ defined by $T(v)=Av$, $A\in \mathrm{Mat}_n(\mathbb R)$. We are given some inner product $\langle ...
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Is continuous function space with standard inner product on $\big[0,\frac{1}{2}\big]$ not complete?

I think Fourier approximation on step function is one example of incompleteness, is it true? Or could you suggest any intuitive examples for incompleteness?
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Orthogonality on complex inner product space

Let $V$ be a complex inner product space. I need to show the following: $(x\ and \ y\ are\ orthogonal)\ \Rightarrow (\left \| \lambda x+\beta y \right \|^{2}=\left | \lambda \right |^{2}\left \| x ...
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On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
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Topology induced by 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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What am I doing wrong? Gram Schmidt process..

Let there be the inner product of all polynomials of degree smaller or equal to 2: $\langle f,g\rangle=\int_0^1f(x)g(x)xdx$. Find orthonormal basis. So I really tried this for an hour and it pretty ...
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Hermitian positive semi-definite matrix is a Gram matrix

I showed that every Gram matrix, i.e. a $n \times n$ matrix $A$ with $A_{ij} = <x_i,x_j>$ where $x_1,...,x_n$ are vectors in an inner product vector space $V$, is Hermitian and positive ...
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Why is this set a half plane?

The set $$\{ x: \| x - x_0\| \leq \|x-a\| , x\in \mathbb{R}^2\}$$ Where we fix the element $a$ and $x_0$ in $\mathbb{R}^2$. I don't see how this is equivalent to $$\{ x: A^t x \leq b\}$$ for some ...
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Does this function define an inner product?

Does the function below define an inner product? $$\langle (x, y), (z, t)\rangle = xz − yt$$ I know how to prove it given two vectors (e.g. $\langle(x,y),(z,t)\rangle$) demonstrating symmetry, ...
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Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
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normalize a vector in $\mathbb C^3$ - a very basic question

I think I forgot a bit previous-year Linear Algebra, so I have a very basic question to you. Given the following question: Normalize the following vector: $v \in \mathbb {C^3}, \space v = i, -i, ...
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What is the relationship between the spaces $\mathscr K (G)$ and $L^2(G)$?

The context is that $G$ is a locally compact Hausdorff group, $\mathscr K (G)$ is the space of continuous compactly-supported functions $G \to \mathbb C$ equipped with the inner product $(f|g) = \int ...
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about vectors norm

in the following article http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf page 3 he say: $$y= \langle y , a_{k_0} \rangle a_{k_0} + R $$ with $a_{k_0}\in D$ with $\forall ...
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Linear form, there exists a unique vector $z$ such that $f(w)=\langle w,z \rangle$

Let $V$ be a finite dimensional space over the field $\mathbb{F}$ with inner product $\langle \cdot, \cdot \rangle$. Then for every linear form $f: V \rightarrow \mathbb{F}$ there exists a unique $z ...
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Kähler form convention

I've been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I'm not missing something. Let's look at this from a purely linear ...
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71 views

Orthogonal complement of a single vector

Is there a quick way to show that if v is an element of an n dimensional real inner product space space, the orthogonal complement of v is n-1 dimensional? I can do this by using gram-schmidt and ...
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Proof that two spans are equal

I have an orthogonal subset of nonzero vectors $u_1,\dots,u_n$. I take $v \in V$ (a vector space) so that $v$ is the in the span of the previous set. Now I let $u$ be $v-m_1u_1-\dots-m_nu_n$ with ...
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70 views

Understanding this inner product

I want to find out under which conditions on $w$, we have that $$\langle f,g \rangle :=\int_0^1 f(x)\bar{g}(x)w(x) dx $$ a dot product?, where $f,g \in C([0,1],\mathbb{C})$ and $w \in ...
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Question about definite positive symmetric bilinear form

Let $A$ be the matrix of a positive definite symmetric bilinear form. Prove $a_{11}a_{nn}\ge a_{1n}a_{n1}$. I don't really have a clue of how to solve this.
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Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
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Inequality involving inner product. $|\langle u,v\rangle+\overline{\langle u,v\rangle}|\le 2|\langle u,v\rangle|$

How is the following inequality involving the inner product true? $|\langle u,v\rangle+\overline{\langle u,v\rangle}|\le 2|\langle u,v\rangle|$ I can see that for a complex number $z=u+iv, ...
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205 views

Determine if the triangle is right-angled using vectors.

Use vectors to decide whether the triangle with vertices $P\langle 1, -3, -2\rangle, Q\langle2, 0, -4\rangle$, and $R\langle6, -2, -5\rangle$ is right-angled. I tried taking the dot product ...
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What does it mean for a vector space to 'have' a particular inner product?

If a vector space 'has' a particular inner product, what does that mean exactly? Is it that all the vectors in the space satisfy the conditions for the inner product to work? Is it that all the ...