2
votes
2answers
58 views

Is $\|x\| = \| \overline{x} \|$ in an inner product space?

Suppose $X$ is a complex inner product space of complex valued functions that is closed under conjugation. Is it true that $\|x\| = \| \overline{x} \|$ for all $x$? If not, is there a simple ...
3
votes
2answers
25 views

Finding the maximum of an integral of a function with given constraints.

This comes from Rudin's Real Analysis text. The first part of the problem asks us to compute $\displaystyle\min_{a,b,c}\int_{-1}^1|x^3-a-bx-cx^2|dx$ (which I have done). Now it asks us to find ...
-1
votes
0answers
28 views

prove of inner product space [on hold]

Show that$$ L^1 (R^n ) $$under the operator $$〈.,.〉:L^1×L^1→R$$ such that $$〈f,g〉=∫_(R^n)▒〖f(x) (g(x)) ̅ 〗$$ for all$$ f,g∈L^1 (R^n )$$forms an inner product space
1
vote
0answers
17 views

How do you show the connection of reproducing kernels to feature maps?

This question is in the context of Hilbert Reproducing Hilbert Spaces and reproducing Kernels and their relation to feature maps (and machine learning). We have a Hilbert space $\mathcal{F}$ and ...
2
votes
1answer
31 views

Inner product in Besicovitch space

Besicovitch space is a space constructed in the following way: We take the closure (with respect to the uniform convergence topology) of a linear span: ...
1
vote
3answers
42 views

Inner product space over generalized number systems

Apologies for the lengthy setup, but I want to make sure I am clear on how I am using the notation, and what I mean by the phrase "generalized number system". Define a generalized number system $G$ ...
1
vote
0answers
16 views

What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
4
votes
1answer
55 views

Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
1
vote
1answer
67 views

What is the difference between an isometric operator and a unitary operator on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is the identity operator) What is the difference between ...
0
votes
1answer
42 views

Sufficient condition for two operators being identical on Hilbert space

Considering two bounded linear operators $S,T$ in $\mathcal{B}(X)$, where $X$ is a complex Hilbert space. If $\def\norm#1#2{\langle {#1},{#2}\rangle} \norm{Sx}{x} = \norm{Tx}{x}$ for all $x\in X$, do ...
0
votes
0answers
26 views

System of equations involving the inner product

I've been reading Ward Cheney's Analysis for Applied Mathematics and he gives the following problem: Indicate how the equation $Ax=b$ can be solved if the operator $A$ is defined by ...
2
votes
0answers
18 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
2
votes
0answers
25 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
0
votes
2answers
36 views

Necessity of hypothesis in distance from a set in an inner product space

In Kreyzig's Functional Analysis book, one of theorems in inner product spaces is about the existence and uniqueness of a minimal point from a set. For lack of better means, I have uploaded the ...
0
votes
0answers
31 views

Underlying concept to this inner-product “quasi-orthogonal-projection”?

I'm looking at a paper about a finite element method for the LLG-equation. [Bartels/Prohl] In the construction of the scheme they are defining a discrete Laplace operator $\tilde{\Delta}_h$ on the ...
0
votes
0answers
21 views

A function $w(t)$ which satisfy $\int dt w(t)F[x](t)=c$ for all x

Consider a differentiable scalar function in two variables $F(x,t)$ for $x\in X$ and $t\in T$, then it can be viewed as an infinite family of scalar functions $\{F[x](t))\}_{x\in X}$. Are there any ...
-1
votes
1answer
42 views

Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...
1
vote
1answer
69 views

Inner product on direct sum of Hilbert spaces

Let $H_1$ and $H_2$ are two different Hilbert spaces then how can we define the inner product on $H_1\oplus H_2$
1
vote
2answers
70 views

Hilbert vs Inner Product Space

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product. Also can a metric always be defined ...
1
vote
0answers
89 views

Duality pairing and difference with inner product in Hilbert spaces

My question is an extension to the post Acting of a dual pairing in Sobolev Spaces. Here duality pairings were discussed and even given explicit examples. Let $U$ and $V$ be Hilbert spaces such that ...
2
votes
1answer
61 views

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 ...
4
votes
3answers
114 views

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 ...
1
vote
3answers
79 views

Prove a space is Hilbert [duplicate]

I got stucked in this problem and get no clue to solve this. Can any one please help me? Thanks Suppose $X$ is an inner product space. If for every bounded linear function $f$, there exists $z \in ...
2
votes
1answer
61 views

Norm of the dual of the Tensor product of Hilbert spaces

Let $V$ and $W$ be Hilbert spaces, we can define inner product and induced norm on Tensor product of these spaces as: Let $v_1,v_2 \in V$,and $w_1,w_2 \in W$. then $(v_1 \otimes w_1, v_2 \otimes ...
3
votes
1answer
33 views

name of matrix of inner products $\langle f_i, f_j\rangle$

Given a Hilbert space $H$ and a number of elements $\phi_i\in H$, does the matrix $M$ with $$ M_{i,j} := \langle\phi_i, \phi_j\rangle $$ have any particular name?
2
votes
1answer
91 views

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 ...
0
votes
0answers
20 views

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 ...
0
votes
1answer
45 views

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 ...
0
votes
4answers
106 views

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 ...
0
votes
1answer
102 views

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 ...
0
votes
1answer
118 views

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 ...
1
vote
2answers
111 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: ...
2
votes
2answers
96 views

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 ( ...
5
votes
1answer
203 views

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 ...
0
votes
0answers
78 views

Conditions of a constant inner product?

For a defined $f$, define $g$ such that: $\langle f,g \rangle = \int_a^b f(x)g(x)\,dx =cte \neq 0$
3
votes
1answer
120 views

How is the inner product in $H^{-1/2}$ defined?

Since $H^{1/2}$ is a Hilbert space, $H^{-1/2}$ must also be a Hilbert space by the isomorphism of Riesz representation theorem. How is the inner product defined there? We know there is a nice ...
0
votes
1answer
51 views

Proof that restriction of hermitian operator to its invariant subspace is also hermitian

Proof that restriction of hermitian operator to its invariant subspace is also hermitian What would be the most elegant way to prove this?
2
votes
1answer
204 views

Continuity of scalar product

In a Hilbert space $H$ with inner product and associated norm, why would if $\|x-x_n\| \longrightarrow 0$ and $\|y-y_n\| \longrightarrow 0$ also $\langle x_n,y_n\rangle \longrightarrow\langle ...
15
votes
4answers
635 views

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 ...
2
votes
1answer
122 views

if $E^2=E$, then $\text{Im}\;E\subset\left(\ker E+(\ker E)^\perp\right)$?

Notation: $V$ is a infinite-dimensional inner product space; $\langle\cdot,\cdot\rangle$ is the inner product of $V$; $E:V\rightarrow V$ is a linear map; $\text{Im}=\{E(v):v\in V\}$; $\ker E=\{z\in ...
3
votes
2answers
61 views

Closed linear subset of a Hilbert space

If $H$ is a Hilbert space, and if $$(a,b)_H=0$$ for every $b \in B \subset H$, where $B$ is a closed linear subset of $H$, does it follow that $a=0$, the zero element of $H$?
1
vote
2answers
239 views

Projection operator for non-orthonormal basis

Let $V \subset H$ be Hilbert spaces. Let $\{v_j\}_{j=1}^\infty$ be a basis for $V$ and $H$. Define $V_N$ to be the span of $\{v_j\}_{j=1}^N$. We can define a projection operator $P:H \to V_N$ by ...
3
votes
1answer
189 views

Proving Inner Product Space

Let $E=C^1 [a,b]$ be the space of all continuously differentiable functions. For $f,g \in E$ define $$ \langle f,g \rangle \ = \ \int_a^b f'(x) \ g'(x) \ dx$$ Is $\langle f,g \rangle$ an inner product ...
1
vote
0answers
37 views

a question in projection

Let $V=L^2(\Omega)$, and $$k=\{v \in V ~s.t ~||v||_{L^2(\Omega)}\leq 1 \}$$ I need to find projection for any $u \in V$ on $k$. Please help me.I do not have any idea about this problem. I have many ...
1
vote
1answer
642 views

Show that Y is a closed subspace of l2

This might be a straight forward problem but I wouldn't ask if I knew how to continue. Apologies in advance, I am not sure how to use the mathematical formatting. We are currently busy with inner ...
1
vote
1answer
278 views

Prove a non-empty subset is closed in an inner product space

I hope someone would be able to help me with the finer details of this proof. Problem: M is a non-empty set in an Inner Product Space (IPS) X. I need to show that the annihilator of M which is ...
2
votes
1answer
35 views

Quick question about sum of subspaces of a Hilbert space

I just have a quick question. Suppose there is $Z$, a Hilbert space, with $A$ and $B$ closed linear subspaces. If $(a,b)=0$ for all $a \in A$ and $b \in B$, I know that $A+B$ is also closed. I don't ...
2
votes
1answer
95 views

inner product space and injective -surjective

Let $V$ and $W$ be two finite-dimensional inner product spaces over the same field and let $T\in \mathcal{L}(V,W)\ $ be a linear transformation. Show that $T$ is injective if $T^*$ is surjective.
2
votes
1answer
91 views

Two inequalities related to norm

We have some difficulties in the following problem: Let $H$ be a real Hilbert space. Find $\alpha>0$ such that $$ \langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq ...
4
votes
2answers
705 views

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 ...