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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Dual of Hilbert space : induced norm vs. operator norm

Let $\mathfrak{H}$ be a Hilbert space. Is the operator norm on the dual $\mathfrak{H}^*$ induced by a inner product ?
M.G
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Getting the unique element in the Riesz-Frechet Theorem.

I have this thorem in my book, H', denotes the dual space, that is the set of bounded linear operators from X to the field over X. The way they got the unique element seems very interesting. Does it follow from the proof of the theorem, that if z…
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Finding rotated orthogonal vectors without knowing lengths

I have two abstract orthogonal vectors $\mid a\rangle$ and $\mid b\rangle$: $\langle a\mid b\rangle=0$, but I don't know the lengths $\mid a\mid=\sqrt{\langle a\mid a\rangle}$ and $\mid b\mid=\sqrt{\langle b\mid b\rangle}$. I would like to find two…
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Determining dimension of a sum of subspaces in terms of a parameter

Problem: Consider the linear subspaces \begin{align*} U = \text{span} \left\{ (1,0,1,0), (1,a,0,a)\right\} \quad \text{and} \quad W = \text{span}\left\{(-1, a, a^2, 0), (0,1,0,-1)\right\} \end{align*} in $\mathbb{R}^4$. Determine $\dim(U^{\bot} +…
Kamil
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The spectrum of a polynomial of an operator, question about proof, why are the operators invertible?

I have a question about a proof. In the proof $\sigma(T)$ is $\{\lambda \in\mathbb{C}: T-\lambda I\text{ is not invertible}\}$. In the proof they use this lemma: Here is the proof, my problem is with the red equivalence: By Lemma 4.35, the…
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Prove there's a unitary linear operator

Let $u, v\in V$, where $V$ is a finite dimensional vector-space, such that $\|u\|=\|v\|$. Prove there's a unitary linear operator such that $T(u) = v$ So if there's such unitary linear operator, it must be that: $$\|u\| = \|T(u)\| = \|v\| =…
jmiller
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if $E,F$, two bases are orthonormal then $T$ is unitary.

Let $T:V\to V$ and two bases of $V$: $E = \{v_1, \ldots, v_n \}$ and $F = \{T(v_1), \ldots, T(v_n)\}$. Prove: $E,F$ are orthonormal implies $T$ is unitary. So basically we want to prove that for every $v\in V$:$$TT^*(v) = 1_v \iff TT^*(v)-1_v(v)…
jmiller
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Normalizing a basis

Let the basis $B = \{1,x,x^2\}$ which is orthogonal. Now, I've seen the following: $$\|1\| = \sqrt {\langle 1,1\rangle} = \sqrt {4\cdot 1\cdot 1} = 2 $$ $$\|x\| = \sqrt {\langle x,x\rangle} = \sqrt {2\cdot 1\cdot 1} = \sqrt 2 $$ $$\|x^2\| =…
jmiller
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Topology property of inner product and norm

It is known that the norm can induce an inner product if and only if it satisfies Parallelogram law. I just want to know what topology property the inner product has while the norm doesn't have?
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Orthogonal basis for $P_{2}(\mathbb{R})$

Consider $P_{2}(\mathbb{R})$ together with inner product: $$\langle p(x), q(x)\rangle = \int_{0}^{1} p(x)q(x) \, dx$$ I am trying to come up with an orthogonal basis with respect to this inner product for $$p(x) = a + bx + cx^2$$ Let $\alpha = \{…
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Proving that $T:\mathbb R^N \rightarrow \mathbb R^N$ is not surjective

Let $T:\Bbb R^n \rightarrow \Bbb R^n$ be a linear transformation and let $u_1,u_2$ different vectors in $R^n$ such that for every $v \in \Bbb R^n$ $Tu_1 \cdot v = Tu_2 \cdot v$. Prove that $T$ is not surjective. I'm clueless about where to begin,…
FigureItOut
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Eigenvalues of Inner Product Matrix

Matrices have a least two major functions in linear algebra. On one hand, they can represent linear transformations as elements of $\text{Hom}(V,V)$). On the other hand they can represent inner products as elements of $\text{Hom}(V,V^*).$ In the…
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Similarity of real symmetric matrices

I've thought about this question for about an hour but I'm still not able to arrive at correct answer. Can anyone suggest me how to go about it?
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Linear algebra: from the inner product to the dot product

Let $W$ be a $n$-dimensional vector space. What do we mean when we say that a linear transformation $f:W\to \mathbb{R}^n$ carries the inner product on $W$ to the dot product on $\mathbb{R}^n$? Is there such a linear transformation? Can you give me…
user249031
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How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$

How do I show that in a normed space $| (\|x\|-\|y\|) | \leq \|x-y\|$ I need to use normed space axioms but I'm unable to figure it out. I think the scalar axiom and triangle inequality are helpful.
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