# Questions tagged [hamel-basis]

A Hamel basis (often simply called basis) of a vector space $V$ over a field $F$ is a linearly independent subset of $V$ that spans it.

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### Finite Dimensional Subspace of a Normed Linear Space is complete.

I am working through the wiki proof that every finite dimensional subspace of a normed linear space is closed. The proof goes as follows: Let $(V ,\| \cdot \|)$ be a normed linear space. Let $W$ be a ...
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### Find subset of vectors which form basis

Question Let W be the subspace of $R^5$ spanned by$u_1 = (1, 2, –1, 3, 4)\\ u_2 = (2, 4, –2, 6, 8) \\ u_3 = (1, 3, 2, 2, 6)\\ u_4 = (1, 4, 5, 1, 8)\\u_5 = (2, 7, 3, 3, 9)$ Find a subset of the ...
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### Does the origin have a strictly convex bounded neighborhood?

Given any finite-dimensional normed space, the topology is equivalent to that generated by the Euclidean norm. So the Euclidean open ball is norm closed. The fact that is is strictly convex is ...
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I am trying to find the reproducing kernel Hilbert space induced by the symmetric positive definite (and bounded and measurable) kernel $$k \colon X \times X \to \{ 0, 1 \}, \qquad (x, y) \mapsto %\... 6 votes 3 answers 688 views ### Is there an infinite dimensional inner product space without an orthogonal Hamel basis? I want to know if there exists an (real or complex) vector space X with infinite dimension and an inner product \langle\cdot,\cdot\rangle such that there is no orthogonal Hamel (algebraic) basis ... 0 votes 0 answers 60 views ### Is the alternative proof correct? if not why? (If \ker f\subset \ker g where f,g  are non-zero linear functionals then show that f=cg for some c\in F.) I came across such a task in the textbook only with the condition if kerf= kerg V = ... 0 votes 1 answer 41 views ### Can any spanning subset be reduced to a Hamel basis in infinite dimensions? As per this question, it is possible to extend any l.i. subset of vectors A \subset V of a vector space to a (Hamel) basis, B \subset V. Is it likewise possible to delete vectors from a spanning ... 2 votes 0 answers 104 views ### On the "basis" of the space C^\infty(a,b) of smooth functions My math methods textbook (Hassani's Mathematical Physics) makes the following (I think dubious) claim: If we assume that a < 0 < b, then the set of monomials 1,x,x^2,... forms a basis for C^∞... 0 votes 1 answer 57 views ### Immediate consequence of a theorem related to spanning set/basis - linear algebra I am going through a theorem and its corollary from a text book: Theorem T_0: Let W be a subspace of R^n, and let B = \{v_1,...,v_p\} be a spanning set of W containing p vectors. Then any ... 1 vote 1 answer 74 views ### Special orthonormal basis for space of continuous real functions on a closed interval Let a,b \in \mathbb{R} with a < b. Let also P = \{x_1, \ldots, x_n\} (with n > 1) be a finite subset of [a,b] with all distinct elements (x_1 < x_2 < \cdots < x_n). Let C[a,... 1 vote 1 answer 38 views ### Fundamental polynomials: do they form a base First, is P_n n or rather n+1 dimensional real vector space of polynomials of degree at most n ? Here, n indicates that it should be n but the basis 1,x,x^2,x^3,...,x^n has n+1 ... 3 votes 1 answer 63 views ### A proof that the null linear mapping is the only one whose matrix representation does not depend on the basis [FALSE] I would like to show the fact that the linear mapping$$ L : E\to E  x\mapsto0_{E} $$is the unique linear mapping whose matrix representation does not depend on the choice of the basis. My ... 3 votes 1 answer 83 views ### Basis criterion for vector space I would like to show the following theorem : Let E be a vector space and S=\{s_i : i\in I\}\subset E. Then we have the equivalence S is a basis of E For any function f : S\to E there exists ... 4 votes 1 answer 136 views ### Prove that the vector space of real convergent power series does not have a countable basis Define V as the subset of \mathbb{R}[[x]], the \mathbb{R}-vector space of formal power series with real coefficients, such that for any f\in V, for any r\in\mathbb{R}, the series f always ... 0 votes 2 answers 57 views ### Find basis of intersection of 2 spans with unequal dimensions I've been stuck on this question for quite a while: Given U = span \left\{ \begin{pmatrix} 0\\ 2\\ 0\\ 0 \end{pmatrix}, \begin{pmatrix} 1\\ 0\\ 0\\ 0 \end{pmatrix}, \begin{pmatrix} 2\\ 1\\ 3\\ 7 \... 1 vote 0 answers 32 views ### Can a infinite dimensional Banach space have a dense Hamel basis [duplicate] I'm looking for a link of a book or paper that has the following Theorem: Let X be an infinite dimensional Banach space. Then there exists a Hamel basis B of X that is dense in X ( That is \... 4 votes 2 answers 143 views ### Why do we prefer the Schauder basis over the Hamel basis in functional analysis? Our functional analysis instructor mentioned in the class that the Hamel basis is not so important in the context of Banach spaces. Instead, we prefer the Schauder basis. However, he did not specify ... 0 votes 0 answers 47 views ### Hamel basis of \ell_p and \ell_\infty. Consider the space \ell_p=\{x=\{x_n\}: \sum|x_n|^p<\infty\} for 1\leq p<\infty.We know that \ell_p can be given \|.\|_q for q>p which is not equivalent to \|.\|_p,the usual norm.... 0 votes 2 answers 468 views ### Proof that a linear map from infinite dimensional vector space to finite dimensional vector space can't be injective In other words, let T: V \to W be an injective linear map. Show that if V is infinite-dimensional then W must also be infinite-dimensional. I think the way I want to approach this is to show ... 0 votes 1 answer 54 views ### Uniqueness of basis of a vector space If X is a vector space and: A \subset X s.t. span(A)=X  ( A basis of X ). If x \in span(A) : x= \sum_{i=1}^{n} x_i k_i for x_1,...,x_n \in X, k_1,...,k_n \in \mathbb{R}. How to prove that ... 1 vote 2 answers 113 views ### Algorithms for extracting a basis Let us consider a rectangle matrix A in M_{mn}(\mathbb{C}) with m<n. Suppose that \operatorname{rank}A=m. What are the well-know algorithms to extract m linearly independent columns of A... 2 votes 0 answers 47 views ### 12 Possible definitions of basis and their properties Let I be any non-empty set, one can consider the set 2^I_{fin} := \{ J \in 2^I \; : \; J \text{ is finite } \} So basically 2^I_{fin} consists of all the finite subset of I. Then the ordered ... 0 votes 0 answers 76 views ### Hamel basis and real algebraic numbers Is the following picture correct Mainly: is it true that a Hamel basis of \mathbb{R} is a subset of \mathbb{R} and that it contains the real algebraic numbers? If so, is there any reference for ... 0 votes 2 answers 263 views ### Understanding the proof of "If V is n dimensional vectors space then a set in V with fewer than n vectors does not span V " In below image author has proved the theorem which states that, "Let V ve a finite dimensional vector space and \{v_1,v_2,...,v_n\} is basis for V then, "If a set in V has fewer than ... 4 votes 1 answer 352 views ### Relation of Hamel basis with the equation f(x + y) = f(x) + f(y)? [duplicate] I am reading "Linear and Nonlinear Functional Analysis with Applications by Philippe G. Ciarlet ", which explains the origin of hamel basis by a problem: Describe the set F of all ... 0 votes 0 answers 35 views ### Where I went wrong in this chain of arguments - Linear algebra Let AB = I_n. A and B are nonsingular, square matrices of size n. Let A_{r1} be the first row of A. The products A_{r1} B_j = 0, j \in \{2,\dots,N\}. B_j is the j^{th} column of B. ... 0 votes 1 answer 83 views ### Total orthonormal set which is not a basis [duplicate] Does there exist a total orthonormal set in a Hilbert space which is not a basis? In a separable Hilbert space every total orthonormal set is a basis. What if the Hilbert space is not separable? 0 votes 1 answer 70 views ### What is an example of spanning set with linearly independent vectors that is not a Hamel Basis? While trying to distinguish the definitions of Hamel and Schauder bases of infinite dimensional vector spaces, I was told that the true definition of a Hamel basis is not "a spanning set with ... 1 vote 0 answers 69 views ### Prove that the algebraic dimension of an infinite dimensional Banach space is atleast \mathfrak{c}. Question: Prove that the algebraic dimension of an infinite dimensional Banach space is atleast \mathfrak{c}. Proof: (X, \|•\|)  be a infinite dimensional Banach space. Then \dim(X) \ge \aleph_{0}... 2 votes 2 answers 51 views ### Given a basis X for a finite dimentional vector space V, prove that L(X) spans L(V). Let L : V \to W be a linear map, and suppose X := \{v_{1},\ldots, v_{n}\} is a basis for V. Prove that Y := \{L(v_{1}),\ldots, L(v_{n})\} is a spanning set for L(V). So I am a struggling on ... 0 votes 1 answer 55 views ### Give a example to show that there not necessarily exist \{e_j\} such that f_i(e_j)=\delta_{ij}, i,j\geq 1. Suppose X is a normed linear space. \{f_i\}\subset X^* are linearly independent. Give a example to show that there not necessarily exist \{e_j\} such that f_i(e_j)=\delta_{ij}, i,j\geq 1. I ... 0 votes 0 answers 187 views ### If S is a spanning set of a vector space, must there be a subset of S which is a basis? Let V be a finite-dimensional vector space. Is it possible that a set S spans V, but no subset of S is a basis of V? I think that the answer is no, at least in the case that S is finite. ... 1 vote 0 answers 74 views ### Difficulty understanding the link between the span of a sequence being an isomorph of \ell_1 and that sequence being a basis of the **closed** span \newcommand{\span}{\operatorname{span}}EDIT: By some further reading, I note that the very fact that \{x_n\} does not contain a weak convergent subsequence implies by Rosenthal's \ell_1 theorem (... 3 votes 0 answers 60 views ### Computability of Vector Representations without Inner Products. Assume you have a separable inner product space V with orthonormal basis B=(e_a)_{a∈A} and an element v∈V. Then there is a simple algorithm to compute the representation of V with respect to ... 0 votes 2 answers 92 views ### Any linear transformation there exists a basis such that \phi(v_i) = \sum_{j=1}^n a_{ij} v_j. Let V be a finite dimensional vector space with basis \{v_1, \cdots, v_n\} over an algebraically closed field K. I want to prove the following theorem. For any linear transformation \phi: V \... 0 votes 1 answer 102 views ### Basis for a column space To find the Col(A) we row reduce the matrix and get R where R is row equivalent to A. Now we choose the columns in A corresponding to the pivot columns in R as the basis. My question is when ... 1 vote 1 answer 31 views ### Linear Independence Condition Let \mathbb{C}^2(\mathbb{R}) be the vector space of all complex two-tuples over \mathbb{R}. I know that the basis is \{(1,0),(i,0),(0,1),(0,i)\}. I also know that showing LINEAR DEPENDENCE of ... 2 votes 2 answers 70 views ### Uncountable subset X of \mathbb{R^{n}} with property that every subset of X with n elements is a basis of \mathbb{R}^{n}. \textbf{Question}: Show that there exist an uncountable subset X of \mathbb{R}^{n} with property that every subset of X with n elements is a basis of \mathbb{R}^{n}. \textbf{My Attempt}: ... 0 votes 2 answers 96 views ### The set of vectors {a, b, c} is a basis of R³. Determine if another set, {3a-2b, a+4b+5c, a-2c}, is also a basis of R³. Since the set of vectors \{\vec{a},\vec{b},\vec{c}\} are a basis of \mathbb{R^3}, we can understand that they are linearly independent. This means that: \alpha_1\cdotp\vec{a}+\alpha_2\cdotp\vec{... 1 vote 0 answers 39 views ### Is a Hilbert space determined by its algebraic dimension? [duplicate] Every Hilbert space is determined up to a unitary isomorphism by the cardinality of its orthonormal basis (which exists by Gram-Schmidt). My question: Let \mathbb{K} be the field either \mathbb{R}... 0 votes 1 answer 265 views ### Basis of a polynomial ring over the field of Rationals I need help checking the validity of the method of my proof and completing it. We observe the polynomial ring \mathbb{Q}[x]  as a vector space over the field \mathbb{Q} it is easy to see from the ... 1 vote 1 answer 46 views ### Boundness of the operator when Hamel basis vectors are eigenvectors Let X be infinite-dimensional Banach space, \{e_j\}_{j\in J} be normalized Hamel basis for it. Suppose for each j\in J we choose a number \lambda_j\in\mathbb{R}. Then we define a linear map$$...
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I don't know whether the answer to this question will help answering this question. Let $V$ be an infinite-dimensional vector space and $A \subset V$ a Hamel basis in $V$. What is the largest (and ...