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Questions tagged [functional-analysis]

Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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Prove $C^1[a,b]$ is a Banach space.

Let $C^1[a,b]$ be the space of continuous differentiable functions on $[a,b]$ equipped with the following norm $$\|x\|=|x(a)| + \sup_{t\in [a,b]}|x'(t)|.$$ Prove that $(C^1[a, b],\|\cdot\|)$ is a Banach space. Here is my attempt: Let $(x_n)$…
Richkent
  • 1,151
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1 answer

completeness of a sequence space

Consider the following space: $X(P):=\{(x_n)_{n\geq0}\in\mathbb{R}^{\mathbb{N}}:(\lambda_nx_n)_{n\geq0}\in l_1, \forall (\lambda_n)_{n\geq0}\in P\}$, where $P$ is a random set of real sequences st. $X(P)$ is separated. Using the familiy…
Mr. Barrrington
  • 871
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1 answer

prove a subset of $l^2$ is closed?

Let $\{f_i\}$ be a sequence of (nice) functions in $L^p[0,1],$ and $p>1, \frac{1}{p}+\frac{1}{q}=1.$ Define a subset $A$ of the space $l^2$ as $$A=\left\{(a_1, a_2, \ldots)\in l^2: \text{ such that } a_i=\int_0^2g(x)f_i(x)dx, \text{ for }g\in…
user108871
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2 answers

About the range of an operator and its adjoint

I need a hand with the proof of this result: If we have an operator between Banach spaces $T:X\to Y$, with closed range, then the adjoint operator $T^*:Y^*\to X^*$ has also closed range. Thanks in advance for any help.
Mark_Hoffman
  • 1,509
2
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2 answers

A doubt in the proof of Uniform Boundedness Theorem as given in Kreyszig.

I was looking up the proof of Uniform Boundedness Theorem. After having proved that the Banach space $X$ is the countable union of closed sets $\bigcup\limits_{i=1} A_i$, and hence satisfies the Baire category theorem, Kreyszig says Some set…
user67803
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1 answer

Properties of a linear operator given an inequality

Let $X$ be a Banach** space and let $X'$ be its dual. Let $x_0 \in X$ and assume that there is $L \in X'$ such that for every $x \in X$ $$\frac{1}{2}\|x_0\|_X^2 - L(x_0) \le \frac{1}{2}\|x\|_X^2 - L(x).$$ I want to prove that $L(x_0) = \|x_0\|_X^2$…
user67133
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0 answers

How to show that the canonical double-dual map $c_0\to\ell^\infty$ is not surjective?

I have a question regarding dual spaces. We know that the dual space of $\ell^1$ is isomorphic to $\ell^\infty$, and the dual space of $c_0$ is isomorphic to $\ell^1$. Here $c_0$ refers to the normed space of sequences of (real or complex) numbers…
Novice
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2 answers

A proof that $T^*$ injective with closed range implies $T$ is surjective

Given $T\in B(H)$ for some Hilbert space $H$, if $T^*$ is injective and has closed range, then $T$ is surjective. My professor sketched a proof by saying that, since $T^*$ has an inverse on its range (by the open mapping theorem), then $T$ maps…
Krystal E.
  • 767
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1 answer

Coercive operators are invertible

Let $V$ be a Hilbert space and let $A:V \to V^*$ be a bounded linear operator such that $$\langle Av, v \rangle \geq C|v|_V$$ for all $v \in V$. Why does this mean that $A$ is an isomorphism $A^{-1}:V^* \to V$ is continuous?? For 1), one can show…
matt.x
  • 605
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1 answer

Is $L^2_{\text{loc}}$ a Hilbert space?

My question is in the title. I know that $L^2$ is a Hilbert space, but I'm not sure about $L^2_{\text{loc}}$. Is it even an inner product space? Thanks.
user121031
2
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1 answer

How to prove that $H-M$ has at least one interior point, provided that $H$ and $M$ are both convex and at least one of them has an interior point?

In the proof of Extended Hyperplane Separation Theorem in Peter. D. Lax's book, there is a statement without proof that since $H$ and $M$ are disjoint convex subsets of linear space $X$, and at least one of them has an interior point, so does $K = H…
zeno tsang
  • 571
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linear map of bounded sets into bounded sets implies a bounded operator

I was watching a video lecture on bounded linear operators from one normed linear space to another. It was stated that if $T$ sends bounded sets in $X$ to bounded sets in $Y$ then $T$ is a bounded operator. I found this to be very hard to prove. The…
harajm
  • 2,117
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1 answer

Inverse of an operator

Let $T:\mathscr{D}(T) \rightarrow H$ be a operator on a complex Hilbert space. If for some $\lambda$ we have that $S(I-\lambda S)^{-1}$ is a bounded inverse for $T-\lambda$ does it then follow that $S$ is an inverse for $T$?
simon
  • 446
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1 answer

This inclusion is continuous

Let $H$ be the real space of all absolute continous functions $F$ from $[0,1]$ to the reals such that $f(0)=0$. I'd like to prove that there is $K>0$ such that for all $f\in H$ $\int_0^1f(x)^2dx\leq K\int_0^1f'(x)^2dx$ Any hint is welcome.
Math.mx
  • 1,919
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1 answer

Riesz representation

Let $H=l^2(N\cup \{0\})$. a. Show that if $\{\alpha_n\}\in l^2$, then the power series $\sum_{n=0}^\infty \alpha_nz^n$ has radius of convergence $\geq 1$. b. If $|\lambda|< 1$ and $L:H\to C $ is defined by $L(\{\alpha_n\}) = \sum_{n=0}^\infty…
saeed
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