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Andrew Stacey
Andrew Stacey
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I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are:

  1. The family of monomials $\{1,t,t^2,t^3,\dots\}$ is a topological basis but not a Schauder basis in $C([0,1])$ because there's not a unique choice of coefficients converging to a given continuous function. (An example I thought of: there's a polynomial approximation of $|t|$ on $[-1,1]$ using only even polynomials, but then that's a polynomial approximation of $t$ on $[-1,1]$$[0,1]$ with zero $t$ coefficient.)

  2. Ditto for trigonometric polynomials.

The reason I ask is because when searching for this on the internet, I came across a statement claiming that trigonometric polynomials weren't a Schauder basis because in general (there's that phrase again!) Fourier series don't converge uniformly for continuous functions. That seems to me like a load of dingo's kidneys (not the convergence statement, but the deduction from it) but - and here's the clincher - the statement was made by someone whose answers on MO I've found to be generally reliable. (To be clear, the statement wasn't made on MO and was somewhere fairly obscure and I'm not going to "name and shame" because I don't want to embarrass that person - if I'm right - or myself - if I'm wrong.)

I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are:

  1. The family of monomials $\{1,t,t^2,t^3,\dots\}$ is a topological basis but not a Schauder basis in $C([0,1])$ because there's not a unique choice of coefficients converging to a given continuous function. (An example I thought of: there's a polynomial approximation of $|t|$ on $[-1,1]$ using only even polynomials, but then that's a polynomial approximation of $t$ on $[-1,1]$ with zero $t$ coefficient.)

  2. Ditto for trigonometric polynomials.

The reason I ask is because when searching for this on the internet, I came across a statement claiming that trigonometric polynomials weren't a Schauder basis because in general (there's that phrase again!) Fourier series don't converge uniformly for continuous functions. That seems to me like a load of dingo's kidneys (not the convergence statement, but the deduction from it) but - and here's the clincher - the statement was made by someone whose answers on MO I've found to be generally reliable. (To be clear, the statement wasn't made on MO and was somewhere fairly obscure and I'm not going to "name and shame" because I don't want to embarrass that person - if I'm right - or myself - if I'm wrong.)

I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are:

  1. The family of monomials $\{1,t,t^2,t^3,\dots\}$ is a topological basis but not a Schauder basis in $C([0,1])$ because there's not a unique choice of coefficients converging to a given continuous function. (An example I thought of: there's a polynomial approximation of $|t|$ on $[-1,1]$ using only even polynomials, but then that's a polynomial approximation of $t$ on $[0,1]$ with zero $t$ coefficient.)

  2. Ditto for trigonometric polynomials.

The reason I ask is because when searching for this on the internet, I came across a statement claiming that trigonometric polynomials weren't a Schauder basis because in general (there's that phrase again!) Fourier series don't converge uniformly for continuous functions. That seems to me like a load of dingo's kidneys (not the convergence statement, but the deduction from it) but - and here's the clincher - the statement was made by someone whose answers on MO I've found to be generally reliable. (To be clear, the statement wasn't made on MO and was somewhere fairly obscure and I'm not going to "name and shame" because I don't want to embarrass that person - if I'm right - or myself - if I'm wrong.)

Source Link
Andrew Stacey
Andrew Stacey
  • 26.8k
  • 12
  • 113
  • 187

Question about Schauder bases in C([0,1]).

I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are:

  1. The family of monomials $\{1,t,t^2,t^3,\dots\}$ is a topological basis but not a Schauder basis in $C([0,1])$ because there's not a unique choice of coefficients converging to a given continuous function. (An example I thought of: there's a polynomial approximation of $|t|$ on $[-1,1]$ using only even polynomials, but then that's a polynomial approximation of $t$ on $[-1,1]$ with zero $t$ coefficient.)

  2. Ditto for trigonometric polynomials.

The reason I ask is because when searching for this on the internet, I came across a statement claiming that trigonometric polynomials weren't a Schauder basis because in general (there's that phrase again!) Fourier series don't converge uniformly for continuous functions. That seems to me like a load of dingo's kidneys (not the convergence statement, but the deduction from it) but - and here's the clincher - the statement was made by someone whose answers on MO I've found to be generally reliable. (To be clear, the statement wasn't made on MO and was somewhere fairly obscure and I'm not going to "name and shame" because I don't want to embarrass that person - if I'm right - or myself - if I'm wrong.)