2
votes
1answer
51 views

How to show: $\exists$ L such that $X\neq L\oplus L^{\perp}$

$X$ : inner product space $L$ : closed subspace of $X$ How to show: $\exists$ L such that $X\neq L\oplus L^\perp$ let $X= (\text{the set of all finite sequences}, \|\cdot\|_2 )$ $L=\lbrace x\in ...
-1
votes
1answer
37 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 ...
0
votes
1answer
27 views

In Inner Product Space ( not complete), dose a closed linear subspace equal to the the orthogonal complement of its orthogonal complement? [duplicate]

It is apparently that this holds in Hilbert space, but I can not prove this for general inner product space or find a counterexample. (The only not complete inner product space known to me is $L^2$ ...
0
votes
1answer
18 views

$x$ is orthogonal to $y$ iff $\|x+ay\| \geq \|x\|$ where $x \in C$

Show that in an inner product space, $x$ is orthogonal to $y$ iff $\|x+ay\| \geq \|x\|$ where $x \in C$. Proof: LHS: If $x$ is orthogonal to $y$, then $\langle x,y\rangle =0$. Let $a \in \mathbb C$. ...
1
vote
1answer
31 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
0answers
39 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 ...
0
votes
1answer
32 views

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 ...
3
votes
3answers
73 views

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 ...
2
votes
1answer
48 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 ...
2
votes
2answers
56 views

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 ...
3
votes
3answers
76 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 ...
0
votes
1answer
60 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 ...
1
vote
0answers
67 views

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 ...
1
vote
3answers
51 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 ...
0
votes
2answers
86 views

Let $z_1 = x_1 +iy_1 $ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $< z_1 , z_2> = x_1x_2+y_1y_2$

Let $z_1 = x_1 +iy_1$ and $z_2 = x_2 +iy_2$ be two complex numbers. The dot product of $z_1$ and $z_2$ is defined by $\langle z_1 , z_2 \rangle = x_1x_2+y_1y_2$ For non zero $z_1$ and $z_2$ prove ...
-1
votes
2answers
85 views

Let V be an inner product space. If $ x⊥y $, then show that

Let $V$ be an inner product space. If $x_i ⊥ x_j $ when $i\neq j$, then show that $$\Bigg\Vert\sum_{i=0}^n x_i\Bigg\Vert^2\ =\sum_{i=0}^n \Vert x_i \Vert^2. $$
3
votes
1answer
57 views

Fredholm alternative and orthonormal basis

The following question relates to the Fredholm alternative: Let $K:H \rightarrow H$ be a compact linear operator and let $I$ be the identity operator. Notation: $N$ is the nullspace and $R$ is the ...
2
votes
1answer
49 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 ...
0
votes
1answer
69 views

Show that orthogonal complement is trivial

I have this subspace of $C[-1,1]$ with inner product $\langle f,g\rangle = \int_{-1}^1f(x)\cdot \bar g(x)\,dx$: $$ E=\left\{f : \int_{-1}^0f=\int_{0}^1f\right\} $$ need to prove that $E^\bot=\{0\}$
0
votes
1answer
92 views

$l_0$ is all sequences with finitely many non-zero terms. Show $W^\perp=\{y: <x,y>=0, x\in W\}=\{0\}$ where $W = \{x : <x,a>=0\}$.

Consider the inner product space $l_0$ consisting of all infinite sequences of complex numbers with only finitely many non-zero terms, with the inner product of $l^2$ (space of square summable ...
0
votes
2answers
90 views

How to prove Schwarz inequality for Hermitian forms?

I'm trying to do something like the proof of the Schwarz inequality for inner product. If $h(y,y)\neq 0$, then we can take $\alpha=-h(x,y)/h(y,y)$ and calculate $h(x+\alpha y,x+\alpha y)$ which is ...
0
votes
3answers
44 views

Proving that $|\langle x,y\rangle|=\|x\|\|y\|$ iff $x,y$ are linearly dependent

Show: For any vectors $x$ and $y$ in an inner product space $V$ over $\mathbb{F}$, we have $$\left|\langle x,y\rangle\right|=\|x\|\|y\| \iff x,y \ are \ dependent$$ ($\leftarrow$) If $x$ and $y$ ...
0
votes
1answer
108 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 ...
7
votes
1answer
147 views

Does this inner product on $L^1([0,1])$ have a name?

Math people: For $f, g \in L^1([0,1])$, define $$\langle f,g \rangle = \int_0^1 \int_0^1 f(t)g(t')\exp(-|t-t'|)dt'\,dt.$$ Although we don't normally think of $L^1([0,1])$ as an inner product space, ...
0
votes
1answer
49 views

Question on polar decomposition of operators.

Suppose $\tau$ is an operator on a finite dimensional complex inner product space. I'm read the following, If $\rho$ is the unique positive square root of the positive operatore $\tau^*\tau$, then $$ ...
0
votes
1answer
57 views

Is a linear functional on $\mathbb{R}^n$ positive if and only if its Riesz vector is positive?

For the following, I say a vector $v>0$ if each of its coordinates is nonnegative and at least one is positive, and $f$ is positive if $v>0$ implies $f(v)\geq 0$. My question is: Suppose $S$ ...
1
vote
1answer
125 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 ...
0
votes
1answer
146 views

norm induced by inner product and triangle inequality

Let $\langle\cdot,\cdot\rangle$ be a scalar product on a space $X$, and let $\lVert \cdot\rVert$ denote the norm induced by this scalar product. I need to show that for $x,y\in X$, $\lVert ...
3
votes
2answers
72 views

Relations between $\|x+a\|$ and $\|x-a\|$ in a normed linear space.

1) Can it happen that $\|x+a\|=\|x-a\|=\|x\|+\|a\|$ when $a\ne0$? 2) How large can $\min(\|x+a\|,\|x-a\|)/\|x\|$ be when $\|x\|\ge \|a\|$? (For a inner-product space, the answers are no and ...
1
vote
2answers
183 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
139 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 ...
0
votes
1answer
195 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 ...
3
votes
3answers
358 views

difference between normed linear space and inner product space

I've seen that the definitions of normed linear space and inner product space for a complex vector space $V$ are very close to each other except for the fact that one is defined on $V$ and the other ...
1
vote
1answer
173 views

Strongly convex function

There is a $\sigma$-strongly convex function, $f(x')\ge f(x)+ \langle x'-x,\mu\rangle +\frac{\sigma}{2}\left|x'-x\right|^2$ where $\mu \in \partial f(x)$, $\mu ' \in \partial f(x')$. How could I get ...
2
votes
1answer
37 views

Inequality in inner product space

Given $V$ an inner product space with norm $(‖v‖_V)^2$=$∫_Ω(v^2 (x)+|∇v|^2 )dx$. Prove that $$(∫_Ω(|v||w|+|∇v||∇w|)dx)^2 ≤ ∫_Ω(|v|^2+|∇v|^2 )dx ∫_Ω(|w|^2+|∇w|^2 )dx=(‖v‖_V)^2(‖w‖_V)^2.$$ Any ...
2
votes
0answers
100 views

Weighted $L^p$ spaces and orthogonal families of polynomials

Let $I$ be a closed interval (bounded or not) of $\mathbb{R}$ and let $w\colon I \rightarrow \mathbb{R}$ be a continuous function that is positive inside $I$ and such that for every $n\in ...
2
votes
1answer
78 views

Self-adjoint operator in the space of twice continuously differentiable functions

There is a problem in the textbook with which I am having difficulties. Prove that operator $A$: $Ay=xy''+y'$ defined on space of twice continuously differentiable functions (scalar product is ...
2
votes
1answer
124 views

Relation between positive definite Hermitian matrices with their inverses

Let $A$ and $B$ be two positive definite Hermitian matrices. Show that the Hermitian matrix $$C\ =\ A^{-1} + B^{-1} - 4(A + B)^{-1}$$ is also positive definite. Thanks in advance.
2
votes
1answer
43 views

Existence of a certain inner product

Does there exist an inner product, such that the all of the monomials $1,x,x^2,\ldots,x^n,\ldots$ (viewed as real valued functions) form an orthogonal (or even orthonormal) set? And what about a ...
1
vote
1answer
117 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
3
votes
2answers
74 views

Does it make sense to talk about $L^2$ inner product of two functions not necessarily in $L^2$?

The $L^2$-inner product of two real functions $f$ and $g$ on a measure space $X$ with respect to the measure $\mu$ is given by $$ \langle f,g\rangle_{L^2} := \int_X fg d\mu, $$ When $f$ and $g$ are ...
0
votes
1answer
59 views

proof that this is an isometric map (on a $C^*$-module)

Are my steps right? I'm not sure about the statement in bold below. Let $A$ be a $C^*$-algebra. Let $X$ be an $A$-module. Let $x\in X$, let $a= \langle x,x \rangle $ Define $\lambda _a (z) = az$, ...
2
votes
1answer
91 views

Self-adjoint operator and inner product

I am wondering whether there is a way to make sense of self adjointness of an operator on $C[0,1]$ without resorting to the inner product of $L^2[0,1]$. I am not referring to concrete alternative ...
0
votes
0answers
71 views

Conditions for a topological vector space to be an inner product space

Are there any conditions on a (real, Hausdorff) topological vector space $\mathcal{V}$ that guarantees the existance of an inner product on that space which induces the same topology? I'm looking for ...
1
vote
0answers
51 views

Determining Similarity of Unit Vectors

I'm seeking for an injective piecewise continuous function $f:\mathbb S^n\rightarrow[0,1]$ where $\mathbb S^N$ is the set of vectors with $L_2$ norm equals $1$. The piecewise continuity requirement ...
3
votes
1answer
114 views

Are the constant functions a closed subspace in the polynomials?

Consider all polynomials $\mathbb R[x]$ and the subspace of polynomials of degree $0$, which we will refer to by the letter $U$. Is this subspace closed with respect to the inner product: $$\langle ...
1
vote
1answer
179 views

Fact about the orthogonal complement of an subset in pre-Hilbert space

I want to show that if $X$ is a pre-Hilbert space and $A$ is a subset of $X$ with an nonempty interior, then $A^{\perp} = \{ 0 \}$. I tried to assume the contrary, then there would be an $x \ne 0$ ...
1
vote
3answers
122 views

Orthogonal projections on a complete convex set

Let assume it is already known that: If $H$ is an inner product space and $\varnothing \neq A \subset H$ is a complete convex subset, then there exists a unique vector $P_A f:=g\in A$ with $\|f-g\| = ...
4
votes
1answer
189 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
184 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: ...