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.

learn more… | top users | synonyms

0
votes
1answer
11 views

Vector notation for sum over elementwise product of 3 vectors

If I have an expression for two vectors $A$ and $B$ as below: $$\displaystyle \sum_{i=1}^N A_i B_i $$ we can write this as $ A^T B $ or $B^T A$ Now, if I have 3 vectors $A$, $B$ and $C$, ...
1
vote
1answer
21 views

minimizing linear combination of inner products

$\mathbf{x},\mathbf{y_1},\mathbf{y_2}\in \mathbb{R}^{m}$ and $\alpha_1,\alpha_2 \in \mathbb{R}$. Also $\|\mathbf{y_1}\|_2 = \|\mathbf{y_2}\|_2 = 1$ and $\alpha_1\geq\alpha_2\geq0$. How should we ...
0
votes
1answer
25 views

Simple question about inner product spaces

$$|\langle x,y\rangle|^2\leq\langle x,x\rangle\langle y,y\rangle$$ Is true in any inner product space, please if someone can show me how to prove the next statement out of the first one ...
0
votes
2answers
20 views

How to find the corresponding matrix of a dot product over a polynomial ring to a specific basis

Let $V= \mathbb R[x]_{\leq 2}$ be the vector-space of real polynomials with degree $\leq 2$. We define a dot product on the $V$ as follows: $$\left<f,g \right> = \int_{0}^1f(x)g(x)dx.$$ ...
-3
votes
0answers
21 views

Candidate inner product [closed]

Consider the space $P_2(\mathbb{R})$ of (real) polynomials of degree two or less with the (candidate) inner product: $$ \langle f,g \rangle = af(0)g(0)+ \int_0^1 f'(t)g'(t) \text{d}t $$ where $f'$ ...
0
votes
1answer
18 views

Does a normed space with norm2 defines an inner product?

I know that generally, an inner product defines a norm on an inner product space, But, generally speaking, If I have a normed space (on purpose I do not say which) with the norm 2 does it mean that I ...
0
votes
2answers
21 views

How to prove that any Hamel basis of an infinite-dimensional complete and separable real inner-product space is uncountable?

How to prove that any Hamel basis of an infinite-dimensional complete and separable (having a countable dense set ) real inner-product space is uncountable ? Do I have to use Baire-category theorem ? ...
1
vote
0answers
27 views

In an infinite dimensional real inner-product space , can any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis?

Let $V$ be an infinite dimensional real inner-product space , then is it true that any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis ? Or at least is it true ...
1
vote
1answer
18 views

Angles in Inner Product Spaces

In inner product spaces, angle is defined to be the only $0 \leq \theta \leq \pi$ satisfies: $$ cos(\theta)=\frac{<v,u>}{||v||\cdot||u||}$$ where $u,v\in V$ - an inner product space. I ...
-2
votes
1answer
38 views

Can there exist a linear operator $T : \mathbb C^2 \to \mathbb C^2$ such that $\langle T(v) , v \rangle =0$ ? [closed]

Can there exist a linear operator $T : \mathbb C^2 \to \mathbb C^2$ such that $\langle T(v) , v \rangle =0$ , where $\langle ., .\rangle$ is the usual inner product over $\mathbb C^2$ ?
3
votes
1answer
41 views

Geometrical or Physical significance (interpretation) of the inner-product $\langle A,B \rangle := Trace (AB^t)$ over $M_n(\mathbb R)$

$\langle A,B \rangle := Trace (AB^t)$ is an inner product over the vector space $M_n(\mathbb R)$ of all real matrices of size $n$ , I would like to know whether this inner-product has any Geometrical ...
2
votes
2answers
54 views

How can I show $U^{\bot \bot}\subseteq \overline{U}$?

Let $H$ be a Hilbert space and $U$ a subspace. Let $U^{\bot}$ denote its orthogonal complement. I had no trouble showing $\overline{U}\subseteq U^{\bot\bot}$. But now I'm stuck for $\supseteq$. ...
0
votes
1answer
45 views

Why is $\langle x-P(x),m\rangle=0$?

Let $H$ be a Hilbert space, and let $M\le H$ be a subspace of it. Let $P:H\rightarrow M$ be the orthogonal projection $H$ onto $M$. We'll take $x\in H$, and $m \in M$. By the definition I know ...
0
votes
1answer
31 views

Question about defintion of inner product space

While practising I came across the following easy question: "Is the space $B(0,1):=\{f:[0,1]\rightarrow\mathbb{R}$ bounded$\}$ an inner product space?" But I'm not quite sure what the correct answer ...
2
votes
1answer
17 views

Expressing the inverse of $C(x) = (\langle x,a_i \rangle )$

Assume we have the following linear transformation: $$C(x) = \tilde{x} = \left( \begin{array}{c} \langle x, a_1 \rangle\\ \vdots \\ \langle x, a_k \rangle\\ \vdots \\ \langle x, a_n \rangle ...
0
votes
2answers
27 views

Prove $\big|\langle x,y \rangle\big| \space ≤ \space \lambda \cdot \|x\|^2+\frac{1}{4\lambda} \cdot \|y\|^2$ in an inner product space

I want to prove that if I have an inner product space with $\lambda>0,$ then $$\big|\langle x,y \rangle\big| \space ≤ \space \lambda \cdot \|x\|^2+\frac{1}{4\lambda} \cdot \|y\|^2$$ Where should I ...
1
vote
1answer
48 views

Expressing $C(x) = \tilde{x} = (\langle x,a_i \rangle )$ as a product of matrices in the form $Cx = \tilde{x}$

Le that $(a_i)^{n}_{i=1}$ be an orthonormal basis and $C(x)$ be a transformation defined as follows: $$C(x) = \tilde{x} = \left( \begin{array}{c} \langle x, a_1 \rangle\\ \vdots \\ \langle x, a_k ...
1
vote
4answers
48 views

A Proof that Orthogonal Complement is unique

So our professor asked us to prove that considering any subspace $S$ of a vector space $V$, the orthogonal complement $S^{\perp}$ is unique. I have devised a proof and I am not sure whether this ...
3
votes
1answer
35 views

Inner-product and vector notation confusion

I am getting increasingly confused with the notation $< , >$ I know the definition of inner-product (simply, dot product) and I also know what a vector is. Does it have two different meanings ...
2
votes
1answer
44 views

Show that the expression does not depend on what base you pick

I got this question on a test today and I had no idea. Define an inner product on the vector space $\mathbb{P}_n$ consisting of polynomials of grade lower than or equal to $n$, by letting $$ \langle ...
1
vote
1answer
19 views

Finding all inner products on $\mathbb{R}^2$ for which a given linear map is orthogonal

Suppose that we are dealing with the real field $\mathbb{R}$ and we define $T:\mathbb R^2\to\mathbb R^2$ by $$T(u,v):=(-v,u).$$ It is trivial to see that $\langle(u,v),T(u,v)\rangle = 0$ for all $u, v ...
2
votes
2answers
50 views

Aren't these linear transformations only orthogonal with respect to *some* inner product?

On p.154 in Husemoller's Fibre Bundles, during his introduction of Clifford algebras, I found a claim which seems questionable to me (highlighted in red): You can click here for some context ...
1
vote
1answer
27 views

Classification of finite dimensional inner product spaces.

Given a complex inner product on a finite-dimensional vector space, is there always a matrix $M$ such that $\langle x,y \rangle=y^*Mx$. What are the properties of such a matrix? I saw on the wiki ...
0
votes
1answer
33 views

Normal endomorphism shares eigenvector with its conjugate and invariant subspace

Given an endomorphism $\alpha$ acting on an inner product space s.t. $\alpha$ is normal, i.e. $\alpha \alpha ^* = \alpha ^* \alpha$, then show there is an eigenvector shared by both $\alpha$ and ...
1
vote
1answer
28 views

question about inner product space

$\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{R}$. given $u\in\mathbb{F}^n$ such as $\langle v,u \rangle=0$ for each $v\in\mathbb{F}^n$. prove that $u=\underline{0}$. by contradiction, ...
1
vote
1answer
36 views

Inner Products on a real two dimensional vector space

Let V be a real two-dimensional vector space with basis $\{e_1, e_2\}$. Find all the inner products $\langle$–,–$\rangle$ on V which satisfy $\langle e_1, e_1 \rangle = \langle e_2, e_2 \rangle = 1$ ...
1
vote
1answer
35 views

Prove that $\det(\text{Id}+T)\ge 1+\det(T)$

Let a self-adjoint operator $T:V\to V$ above $\mathbb{C}$, such that $\langle Tv,v \rangle \ge 0$ (so it's essentially a real number). We have learned before that for this kind of $T$, all it's ...
2
votes
1answer
31 views

Prove a generalization for a formula involving an inner-product

Let $V$, a vector space over $\mathbb{C}$ from a finite dimension with an inner product $\beta \langle .,. \rangle$. Let $B=\{v_1,\ldots, v_n\}$, a basis for $V$. If $v_1,\ldots,v_n$ is an orthonormal ...
2
votes
1answer
35 views

Theorem 3.6-4 in Erwin Kreyszig's Introductory Functional Analysis With Applications

Here's the statement of Theorem 3.6-4 in Erwin Kreyszig's Introductory Functional Analysis With Applications: Let $H$ be a Hilbert space. Then (a) If $H$ is separable, then every orthonormal set in ...
1
vote
2answers
53 views

A condition equivalent to orthogonality

Prove that in any inner product space: $x$ and $y$ are orthogonal if and only if $||x+\alpha y||\ge ||x||$ for every scalar $\alpha$.
1
vote
1answer
25 views

An operator and the product of it with its adjoint have the same kernel

I'm studying for my qualifying exams, and I came across the following question. I want to make sure I'm on the right track with my proof: Suppose $(V, \left\langle , \right \rangle)$ is an inner ...
0
votes
0answers
24 views

$( \alpha \mid \beta)=\operatorname{Re }( \alpha \mid \beta) + i \operatorname{Im} (\alpha \mid \beta)$

A book I am using for an independent study of linear algebra states that if $V$ is a complex vector space with an inner product then for all $\alpha, \beta \in V$ we have $$( \alpha \mid ...
0
votes
1answer
25 views

Describe explicitly all inner products on $\mathbb{R}$ and $\mathbb{C}$

I know this is an elementary question, however I am really lost as to where to start. Since both $\mathbb{R}$ and $\mathbb{C}$ are finite-dimensional I think the inner product will be completely ...
2
votes
2answers
38 views

Convergence in inner product space

Let $X$ be an inner product space. If $x_n\to x$, $y_n\to y$ (in norm), then $(x_n,y_n)\to(x,y)$ (in modulus).
0
votes
1answer
26 views

Modified inner product

Given two real valued orthogonal functions, say $f(x)$ and $g(x)$, if we define an inner product $$ \langle f,g\rangle \ = \ \int_a^b f(x) g(x) dx,$$ which we know satisfies the properties of an ...
3
votes
2answers
51 views

How to prove a Banach normed vector space is NOT a Hilbert space?

We know that the Banach space $\big(\Bbb R^n,\|\cdot\|_2\big)$ is a Hilbert space with inner product $\langle x,y\rangle := \sum_{k=1}^n x_ky_k$. However, how to prove that $\big(\Bbb ...
-1
votes
1answer
58 views

Why is $\langle p,q \rangle = 0$? [duplicate]

First of all, sorry for opening a new question about it, but I'm curious to understand: John Hughes claims that $\langle p,q \rangle =0$ (in the end of his answer) Why is it true? Prove that there ...
0
votes
1answer
49 views

Prove that there is a unique inner product on $V$

Let $V$, a vector space over $\mathbb{F}$ and $W_1,W_2 \subseteq V$ such that $W_1 \oplus W_2 = V$. For $i=1,2$, let $\langle , \rangle $ on $W_i$. Prove that there is a unique inner product on $V$ ...
0
votes
1answer
43 views

Prove/Disprove: $\forall u\in V: \langle v,u \rangle = \langle w,u \rangle \implies v=w$.

Prove/Disprove: $\forall u\in V: \langle v,u \rangle = \langle w,u \rangle \implies v=w$. I want to say "Yes", but couldn't formulate my intuition into a proof. How to prove it?
3
votes
3answers
65 views

Orthogonal complement propertiss

I'm required to proove that $W^{\perp} + U^{\perp} \subseteq (W \cap U)^{\perp}$. I've already proven $U \subseteq W \to W^{\perp} \subseteq U^{\perp}$ and $(U+W)^{\perp} = W^{\perp} \cap U^{\perp}$. ...
1
vote
2answers
35 views

What is the orthogonal complement

Let $\langle A,B \rangle = \text{tr}(B^TA)$. For $M_n(\mathbb{R}$): Find the orthogonal complement of the diagonal-square-matrices. So I need to find $$U^{\perp} = \left\{ A\in M_n(\mathbb{R}) \mid ...
0
votes
1answer
57 views

Given a vector $u$ such that $\langle v,u\rangle=0$ for every vector $v$, prove that $u=0$ [closed]

Suppose $\langle \cdot, \cdot\rangle $ is an inner product on $\mathbb{F}^n$. Given a vector $u$ such that $\langle v,u\rangle=0$ for every vector $v$, prove that $u=0$. Here $\mathbb F$ is a ...
0
votes
1answer
41 views

Why is $ \mathrm{Re}\langle v,iu\rangle=- \mathrm{Im}\langle u,v\rangle$ true?

Why is this identity true? $$\mathrm{Re}\langle v,iu\rangle=- \mathrm{Im}\langle u,v\rangle$$ If we start from LHS: $$\mathrm{Re}\langle v,iu\rangle = \mathrm{Re}\left(-i\langle ...
0
votes
2answers
50 views

Prove an identity for an inner product

Prove that identity $\langle u,v\rangle = \text{Re}\langle v,u \rangle - i\text{Re}\langle v,iu\rangle $ Where $V$ is an inner product over $\mathbb{C}$. I tried to use the common properties of ...
0
votes
1answer
30 views

Must $T:V \to V$ be the $0$ transformation if $\langle T(v),u\rangle=0 \forall u,v \in V$?

Must $T:V \to V$ be the $0$ transformation if $\langle T(v),u\rangle=0 \ \forall u,v \in V$? My intuition say $T$ must be the $0$ transformation, can someone give me a formal proof please?
0
votes
1answer
26 views

Why $\|Ax\|^2=\frac{1}{4}\left(2\left<A(\beta x), v\right>+2\left<Av,\beta x\right>\right)$

Let $A$ a symmetric matrix, $x\in X$ where $\left(X,\left<\cdot \right>\right)$ is a inner product space, $\beta\in\mathbb R\backslash \{0\}$ and $v=\frac{Ax}{\beta}$. My question is probably ...
0
votes
0answers
40 views

What makes a norm-Gaussian inner product space “infinite-dimensional”?

Suppose we define an $\mathbb{R}^m$ inner product space in which the inner product of $\mathbf{x}$ and $\mathbf{y}$ is $\exp\left(-\|\mathbf{x} - \mathbf{y}\|\right)$. In PCA and machine learning, we ...
0
votes
1answer
26 views

Finding uniq inner product satisying the following requierments:

Let $V$ be a vector space over field $\mathbb{F}$, and let $W_1,W_2 \subseteq V$ be sub-spaces such that $V=W_1 \oplus W_2$. Let $\langle,\rangle_i$ be an inner product on $W_i$ for $i=1,2$. Prove ...
0
votes
1answer
21 views

Reference for Weak convergence in hilbert space

I want to understand weak convergence in hilbert space. Can somebody give me a reference for that. Thanks in advance.
2
votes
1answer
34 views

Inner Product on polynomials over field of complex numbers

I am playing with the simplest of polynomial vector spaces - the Legendre polynomials (I hope I have that name right! :-) where $\langle P,Q\rangle = \int_{-1}^{+1}P(x)Q(x)dx$ This is straightforward ...