Tagged Questions
2
votes
1answer
68 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
26 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
54 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
37 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
33 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
39 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 ...
0
votes
1answer
55 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
26 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
48 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
56 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
$$
\langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq
...
2
votes
2answers
105 views
Isomorphisms of inner-product spaces
I think I understand why all finite-dimensional vector spaces over a field $\mathbb{K}$ are isomorphic to $\mathbb{K}^n$. Any linear map $T: V \rightarrow W$ between finite-dimensional vector spaces ...
1
vote
1answer
50 views
Inner product? Yes or no?
I define an "inner product" on $H_0^2(U)$ where $U \subset R^n$ is bounded open set:
$$\langle u,v\rangle = \int_U \Delta u \Delta v dx.$$
I need this when trying to find a weak solution for my PDE ...
1
vote
1answer
90 views
Hilbert spaces and orthogonality sets
I need to prove if $X$ is a Hilbert space and $M$ and $N$ it's closed:
$$
(M+N)^\perp=M^\perp\cap N^\perp
$$
thanks
0
votes
1answer
75 views
Question about L2 Inner Product and Integrals
Does exists $f\in L_2(\mathbb{R}^d)$ such that for all $g\in L_2(\mathbb{R}^d)$ which is not identically zero:
...
4
votes
1answer
88 views
Show that if the Riesz map is surjective on $H$, then $H$ is a Hilbert space
Let $H$ be a vector space equipped with an inner product $(\cdot, \cdot)$ and $f:H\to H',\ f(x)=(\cdot,x)$ surjective. Now, why $H$ is a Hilbert space?
The other direction is clear by Riesz' ...
2
votes
2answers
98 views
Completion of pre-Hilbert space in H. Brezis' Functional Analysis
I'm trying to solve the problem 5.12 of Harim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations; but I'm stucked understanding the statement which comes as follows:
...
6
votes
3answers
153 views
Why isn't it a Hilbert space
Let $X$ be the vector space of all the continuous complex-valued functions on $[0,1]$. Then $X$ has an inner product $$(f,g) = \int_0^1 f(t)\overline{g(t)} dt$$ to make it an inner product space. But ...
2
votes
1answer
56 views
Question about linear operators on Hilbert spaces
I have the following question: "Let $V$ be a Hilbert space and let $T$ be a linear operator on $V$. If $S$ is any linear operator on $V$ that satisfies $\langle Tv,w \rangle = \langle Sv,w \rangle$ ...
2
votes
2answers
88 views
Theorem about orthogonal system in inner product space.
It is known that "If $\{x_n\}$ is a sequence in a real Hilbert space $H$
satisfying
$$
\langle x_n, x_m\rangle =0 \quad\forall n\ne m,
$$ then $\displaystyle\sum_{n=1}^{\infty}x_n$ is convergent if ...
2
votes
1answer
118 views
Finding a linear mapping in a special Hilbert space
Let $H=\ell_2$, the real Hilbert space whose elements are the
square-summable sequences of real scalars, i.e.,
$$
H=\left\{u=(u_1,u_2,\ldots,u_i,\ldots):
...
4
votes
2answers
92 views
Orthogonality checking in Kreyszig exercise
Let $H$ be inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $\lVert \cdot\rVert$. Let $x,y \in H$. Would you help me to prove that $\langle x,y\rangle=0$ if and only if ...
0
votes
0answers
94 views
Set of Bounded linear Operators on $l_2$ is dense on the set of bounded operators on $l_2$?
Let $l_2^{+}$ be the Hilbert space of all square summable sequences $\{x_n\}, n \in \mathbb{N}$ under some definition of inner-product $\langle,\rangle_l$. Define $B[l_2^{+}]$ as the set of all ...
1
vote
1answer
140 views
Representing with Hilbert Schmidt Norm
Am trying to see, if the following Trace function can be expressed using a Hilbert Schmidt Norm: $\operatorname{Tr}(X^TAX)$. Here, $X$ is a matrix whose entries take values that are finite and reals ...
0
votes
0answers
109 views
Dimension of Hilbert spaces obtained by the completion of countably-infinite dimensional inner product spaces
Let $\mathcal{V}$ be a countably infinite dimensional inner product space over $\mathbb{C}$ and Hilbert space $\mathcal{H}$ be the completion of $\mathcal{V}$. Will $\mathcal{H}$ always be a countably ...
2
votes
2answers
89 views
Minimizing a functional on $L^2$
Let
$$
\mathcal{M} := \left\{f \in L^2([0,\pi]): \int_0^\pi f(x)\cos x dx = \int_0^\pi f(x)\sin x dx = 1\right\}.
$$
Solve this problem:
$$ \tag{P}
\min_{\mathcal M} \int_0^\pi ...
1
vote
1answer
122 views
Under What Conditions Does the Action of the Dual Space Induce an Hermitian Inner Product?
I'm starting to learn about Dirac notation in Quantum Mechanics, and am coming from a pure background. The notes I'm reading states that we assume that the action of the dual space on the state space ...
4
votes
1answer
116 views
Convergence of a series involving inner products
Let $\{A_{j}\}$ be a sequence of bounded operators on a Hilbert space satisfying $\|A_{j}^{\ast}A_{k}\| \leq C_{j - k}$ and $\|A_{j}A_{k}^{\ast}\| \leq C_{j - k}$ where $\sum C_{i} < \infty$. Fix an ...
2
votes
1answer
153 views
An orthonormal family in an inner product space
Why does the inner product space $( C[0,1], \| \cdot \|_2$) have an orthonormal family $(e^{\color{red}{2\pi}inx})_{n\in \mathbb{N}}$ ?
2
votes
3answers
138 views
orthonormal basis in $l^{2}$
I need an orthonormal basis in $l^{2}$. One possible choice would be to take as such the sequences $\{1,0,0,0,...\}, \{0,1,0,0,...\}, \{0,0,1,0,...\}$, but I need a basis where only finitely many ...
3
votes
2answers
283 views
A counterexample to theorem about orthogonal projection
Can someone give me an example of noncomplete inner product space $H$, its closed linear subspace of $H_0$ and element $x\in H$ such that there is no orthogonal projection of $x$ on $H_0$. In other ...
5
votes
1answer
122 views
Addition of Unbounded Operators
Let $H$ be a (separable complex) Hilbert space and let $A$ and $B$ be two densely-defined, maximally-defined linear operators on $H$ with domains $D(A)$ and $D(B)$ respectively. (By maximall-defined, ...
7
votes
1answer
614 views
Equivalent inner products on a Hilbert space
Take a Hilbert space $(\mathcal H,(\cdot,\cdot)_{\mathcal H})$ and two equivalent inner products $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$ on $\mathcal H$, i.e. such that there are $a,b \in \mathbb R$ ...
3
votes
0answers
244 views
How does the parallelogram law imply the existence of an inner product for a given norm? [duplicate]
Possible Duplicate:
Norms Induced by Inner Products
I am trying to prove to that if a norm of a vector space satisfies the parallelogram law ($\| \vec x + \vec y \|^2 + \| \vec x - \vec ...
2
votes
3answers
849 views
Canonical examples of inner product spaces that are not Hilbert spaces?
That is, what are some good examples of vector spaces which are inner product spaces but in which not every Cauchy sequence converges?
17
votes
2answers
1k views
Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)
I am trying to understand the differences between
$$
\begin{array}{|l|l|l|}
\textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline
\text{metric}& \text{metric ...
4
votes
1answer
317 views
Summation of inner products
I can't seem to find a way of asking a sub-question in relation to does linearity of inner product hold for infinite sum, which is in itself too generic a question for my purposes. Could someone ...
7
votes
1answer
213 views
Energy estimate of the differential equation $\dot{x}=Ax$
Conside the differential equation
$$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...
0
votes
1answer
169 views
Form of the inner product in $\ l_2$
Is it true that every inner product in $\ l_2 $ is of the form $\langle x,y\rangle_a =\sum_{n=1} ^ {\infty} {a_n x_n y_n}$ ? (Of course $\ x=(x_n) , y=(y_n) $ are in $\ l_2 $ .)
