# 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).

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### Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
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### Discontinuous linear functional

I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, ...
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### Understanding of the theorem that all norms are equivalent in finite dimensional vector spaces

The following is a well-known result in functional analysis: If the vector space $X$ is finite dimensional, all norms are equivalent. Here is the standard proof in one textbook. First, pick a ...
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### How do you show that $l_p \subset l_q$ for $p \leq q$?

I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
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### “Every linear mapping on a finite dimensional space is continuous”

From Wiki Every linear function on a finite-dimensional space is continuous. I was wondering what the domain and codomain of such linear function are? Are they any two topological vector spaces (...
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### Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
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### Example of a closed subspace of a Banach space which is not complemented?

In this post, all vector spaces are assumed to be real or complex. Let $(X, ||\cdot||)$ be a Banach space, $Y \subset X$ a closed subspace. $Y$ is called $\underline{\mathrm{complemented}}$, if there ...
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Statement: If $\lambda$ is an eigenvalue of $AB^{-1}$, then $\lambda$ is an eigenvalue of $B^{-1}A$ and vice versa. One way of the proof. We have $B(B^{-1}A ) B^{-1} = AB^{-1}.$ Assuming $\... 4answers 5k views ### The direct sum of two closed subspace is closed? (Hilbert space) I know that if$X$is a Banach space, then, the direct sum of two closed subspace$X_1$and$X_2$is not necessarily closed. But what if$X$is Hilbert? I assume there is something to do with the ... 3answers 28k views ### Derivative of convolution Assume that$f(x),g(x)$are positive and are in$L^1$. Moreover, they are differentiable and their derivative is integrable. Let$h(x)=f(x)*g(x)$, the convolution of$f$and$g$. Does the derivative ... 2answers 2k views ### Complement of$c_{0}$in$\ell^{\infty}$How can I show that$c_{0}$cannot be complemented in$\ell^{\infty}$? Complement in the following sense $$c_{0}+V = \ell^{\infty}$$ And the projections are continuous. 1answer 3k views ### Compactness of Multiplication Operator on$L^2$Suppose we have an bounded linear operator A that operates from$L^2([a,b]) \mapsto L^2([a,b])$. Now suppose that$A(f)(t) = tf(t)$. Is A compact? Edit: I know$A = A^*$but I'm not really sure ... 2answers 7k views ### Example to prove that$ C^1[0,1] $is not a Banach space for the uniform norm? The space$ C^1[0,1] $- the space of all continuously differentiable functions on$ [0,1]$is not a Banach space with respect to the sup norm,$ \|.\|_{\infty} $since the uniform limit of a ... 1answer 1k views ### Orthogonal complement examples I am looking for an example such that in a pre-Hilbert space$H$we have for a subspace$U$that (i)$\bar{U} \oplus U^\perp \neq H$(ii)$ \bar{U} \neq U^{\perp \perp}$Since finite and closed ... 2answers 2k views ### A net version of dominated convergence? Let$G$be a locally compact Hausdorff Abelian topological group. Let$\mu$be a Haar measure on$G$, i.e. a regular translation invariant measure. Let$f$be fixed in$\mathcal{L}^1(G, \mu)$. ... 1answer 42k views ### Relations between p norms The p-norm is given by$||x||_{p} = (\sum_{n=1}^{\infty }|x_{n}|^{p})^{1/p}$. For$0 < p < q$, it can be shown that$||x||_{p} \geq ||x||_{q}$(1, 2). It appears that in$R^{n}$a number of ... 2answers 7k views ### Compact operator maps weakly convergent sequences into strongly convergent sequences I found the following property of compact operators in a proof, and I can't prove it. Prove that if$T \in \mathcal{L}(E,F)$is compact, and if$u_n \rightharpoonup u$(the sequence converges ... 2answers 3k views ### Intersection of kernels and linear dependence of functionals I am trying to prove the following. I have seen it alluded to in other places of the internet (this site included) but without proof. Let$L,L_1\ldots L_n$be linear functionals on a vector space$X$.... 5answers 4k views ### A natural proof of the Cauchy-Schwarz inequality Most of the proofs of the Cauchy-Schwarz inequality on a pre-Hilbert space use a fact that if a quadratic polynomial with real coefficients takes positive values everywhere on the real line, then its ... 1answer 2k views ### If$1\leq p < \infty$then show that$L^p([0,1])$and$\ell_p$are not topologically isomorphic If$1\leq p < \infty$then show that$L^p([0,1])$and$\ell_p$are not topologically isomorphic unless$p=2$. Maybe I would have to use the Rademacher's functions. 15answers 37k views ### Good book for self study of functional analysis I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic ... 2answers 19k views ### Norm of a symmetric matrix equals spectral radius How do I prove that the norm of a matrix equals the absolutely largest eigenvalue of the matrix? This is the precise question: Let$A$be a symmetric$n \times n$matrix. Consider$A$as an ... 2answers 6k views ###$T$is continuous if and only if$\ker T$is closed Let$X,Y$be normed linear spaces. Let$T: X\to Y$be linear. If$X$is finite dimensional, show that$T$is continuous. If$Y$is finite dimensional, show that$T$is continuous if and only if$\ker ...
If $V$ and $W$ are topological vector spaces (and $W$ is finite-dimensional) then a linear operator $L\colon V\to W$ is continuous if and only if the kernel of $L$ is a closed subspace of $V$. Why is ...